Research Article Fuzzy Logic for Incidence Geometrydownloads.hindawi.com/journals/tswj/2016/9057263.pdf · ] we will use the following axioms from [ ]. es e axioms formalize the behaviour

Research ArticleFuzzy Logic for Incidence Geometry

Alex Tserkovny

Dassault Systemes 175 Wyman Street Waltham MA 02451 USA

Correspondence should be addressed to Alex Tserkovny atserkovnyyahoocom

Received 3 November 2015 Revised 2 June 2016 Accepted 26 June 2016

Academic Editor Oleg H Huseynov

Copyright copy 2016 Alex Tserkovny This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The paper presents a mathematical framework for approximate geometric reasoning with extended objects in the context ofGeography in which all entities and their relationships are described by human language These entities could be labelled bycommonly used names of landmarks water areas and so forth Unlike single points that are given in Cartesian coordinates thesegeographic entities are extended in space and often loosely defined but people easily perform spatial reasoning with extendedgeographic objects ldquoas if they were pointsrdquo Unfortunately up to date geographic information systems (GIS) miss the capability ofgeometric reasoning with extended objectsThe aim of the paper is to present a mathematical apparatus for approximate geometricreasoning with extended objects that is usable in GIS In the paper we discuss the fuzzy logic (Aliev and Tserkovny 2011) as areasoning system for geometry of extended objects as well as a basis for fuzzification of the axioms of incidence geometryThe samefuzzy logic was used for fuzzification of Euclidrsquos first postulate Fuzzy equivalence relation ldquoextended lines samenessrdquo is introducedFor its approximation we also utilize a fuzzy conditional inference which is based on proposed fuzzy ldquodegree of indiscernibilityrdquoand ldquodiscernibility measurerdquo of extended points

1 Introduction

In [1ndash4] it wasmentioned that there are numerous approachesby mathematicians to restore Euclidean Geometry from adifferent set of axioms based on primitives that have exten-sion in space An approach aimed at augmenting existentaxiomatization of Euclidean geometry with grades of validityfor axioms (fuzzification) is also presented in [1ndash4] But incontrast with [1ndash4] where theLukasiewicz logicwas only pro-posed as the basis for ldquofuzzificationrdquo of axioms and no proofswere presented for both fuzzy predicates and fuzzy axiom-atization of incidence geometry we use fuzzy logic from[5] for all necessary mathematical purposes to fill up above-mentioned ldquogaprdquo

2 Axiomatic Geometry and Extended Objects

21 Geometric Primitives and Incidence Similarly to [1ndash3 6ndash8] we will use the following axioms from [9] These axiomsformalize the behaviour of points and lines in incidentgeometry as it was defined in [1]

(I1) For every two distinct points p and q at least one line lexists that is incident with p and q

(I2) Such a line is unique(I3) Every line is incident with at least two points(I4) At least three points exist that are not incident with the

same line

The uniqueness axiom (I2) ensures that geometrical con-structions are possible Geometric constructions are sequen-tial applications of construction operators An example of aconstruction operator is the following

Connect point times pointrarr line

Take two points as an input and return the line throughthem For connect to be a well-defined mathematical func-tion the resulting line needs always to exist and needs to beunique Other examples of geometric construction operatorsof 2D incidence geometry are the following

Intersect line times linerarr pointParallel through point line times pointrarr line

Theaxioms of incidence geometry formaproper subset ofthe axioms of Euclidean geometry Incidence geometry allows

Hindawi Publishing Corporatione Scientific World JournalVolume 2016 Article ID 9057263 18 pageshttpdxdoiorg10115520169057263

2 The Scientific World Journal

for defining the notion of parallelism of two lines as a derivedconcept but does not permit expressing betweenness or con-gruency relations which are assumed primitives in Hilbertrsquossystem [9] The complete axiom set of Euclidean geometryprovides a greater number of construction operators thanincidence geometry Incidence geometry has very limitedexpressive power when compared with the full axiom system

The combined incidence axioms (I1) and (I2) state that itis always possible to connect two distinct points by a uniqueline In case of coordinate points a and b Cartesian geometryprovides a formula for constructing this unique line

119897 = 119886 + 119905 (119887 minus 119886) | 119905 isin 119877 (1)

As it was shown in [1ndash4] when we want to connect twoextended geographic objects in a similar way there is nocanonical way of doing so We cannot refer to an existingmodel like the Cartesian algebra Instead a new way of inter-preting geometric primitives must be found such that theinterpretation of the incidence relation respects the unique-ness property (I2)

Similarly to [1ndash4] we will refer to extended objects thatplay the geometric role of points and lines by extendedpoints and extended lines respectively Section 3 gives a briefintroduction on proposed fuzzy logic and discusses possibleinterpretations of fuzzy predicates for extended geometricprimitives The fuzzy logic from [5] is introduced as pos-sible formalism for approximate geometric reasoning withextended objects and based on extended geometric primitivesfuzzification of the incidence axioms (I1)ndash(I4) is investigated

3 Fuzzification of Incidence Geometry

31 Proposed Fuzzy Logic Let forall119901 119902 isin [0 1] and continuousfunction (119901 119902) = 119901 minus 119902 which defines a distance between 119901and 119902 Notice that 119865(119901 119902) isin [minus1 1] where 119865(119901 119902)min

= minus1

and 119865(119901 119902)max= 1 When normalized the value of 119865(119901 119902) is

defined as follows

119865 (119901 119902)norm

=

119865 (119901 119902) minus 119865 (119901 119902)min

119865 (119901 119902)max

minus 119865 (119901 119902)min

=

119865 (119901 119902) + 1

2

=

119901 minus 119902 + 1

2

(2)

It is clear that 119865(119901 119902)norm isin [0 1] This function repre-sents the value of ldquoclosenessrdquo between two values (potentiallyantecedent and consequent) defined within single intervalwhich therefore could play significant role in formulation ofan implication operator in a fuzzy logic Before proving that119868(119901 119902) is defined as

119868 (119901 119902) =

1 minus 119865 (119901 119902)norm

119901 gt 119902

1 119901 le 119902

(3)

and 119865(119901 119902)norm is from (2) let us show somebasic operations

in proposed fuzzy logic Let us designate the truth valuesof logical antecedent 119875 and consequent 119876 as 119879(119875) = 119901 and119879(119876) = 119902 respectively Then relevant set of proposed fuzzy

Table 1

Name Designation ValueTautology 119875

119868 1Controversy 119875

119874 0Negation not119875 1 minus 119875

Disjunction 119875 or 119876

119901 + 119902

2

119901 + 119902 lt 1

1 119901 + 119902 ge 1

Conjunction 119875 and 119876

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

Implication 119875 rarr 119876

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 le 119902

Equivalence 119875 harr 119876

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

Pierce arrow 119875 darr 119876

1 minus

119901 + 119902

2

119901 + 119902 lt 1

0 119901 + 119902 ge 1

Shaffer stroke 119875 uarr 119876

1 minus

119901 + 119902

2

119901 + 119902 gt 1

1 119901 + 119902 le 1

logic operators is shown in Table 1 To get the truth values ofthese definitions we use well-known logical properties suchas 119901 rarr 119902 = not119901 or 119902 119901 and 119902 = not(not119901 or not119902) and the like

In other words in [5] we proposed a new many-valuedsystem characterized by the set of base union (cup) andintersection (cap) operations with relevant complement definedas 119879(not119875) = 1 minus 119879(119875) In addition the operators darr and uarr areexpressed as negations of the cup and cap correspondingly Forthis matter let us pose the problem very explicitly

We areworking inmany-valued systemwhich for presentpurposes is all or some of the real intervalR = [0 1] As wasmentioned in [5] the rationales there aremore than ample forall practical purposes the following set 0 01 02 09 1of 11 values is quite sufficient and we will use this set 119881

11in

our illustration Table 2 shows the operation implication inproposed fuzzy logic

32 Geometric Primitives as Fuzzy Predicates It is well knownthat in Boolean predicate logic atomic statements are formal-ized by predicates Predicates that are used in the theory ofincidence geometry may be denoted by 119901(119886) (ldquoa is a pointrdquo)119897(119886) (ldquoa is a linerdquo) and inc(119886 119887) (ldquoa and b are incidentrdquo) Thepredicate expressing equality can be denoted by eq(119886 119887) (ldquoaand b are equalrdquo) Traditionally predicates are interpreted bycrisp relations For example eq119873times119873 rarr 0 1 is a functionthat assigns 1 to every pair of equal objects and 0 to everypair of distinct objects from the set N Of course predicates

The Scientific World Journal 3

ch(A)

c(A)

A

120579max(A)

120579min(A)

(a)

A

B120579max(ch(A cup B))

120579max(B)

120579max(A)

(b)

Figure 1 (a) Minimal and maximal diameter of a set A of Cartesian points (b) Grade of distinctness 119889119888(119860 119861) of A and B

Table 2

119901 rarr 119902 0 01 02 03 04 05 06 07 08 09 10 1 1 1 1 1 1 1 1 1 1 101 045 1 1 1 1 1 1 1 1 1 102 04 045 1 1 1 1 1 1 1 1 103 035 04 045 1 1 1 1 1 1 1 104 03 035 04 045 1 1 1 1 1 1 105 025 03 035 04 045 1 1 1 1 1 106 02 025 03 035 04 045 1 1 1 1 107 015 02 025 03 035 04 045 1 1 1 108 01 015 02 025 03 035 04 045 1 1 109 005 01 015 02 025 03 035 04 045 1 11 0 005 01 015 02 025 03 035 04 045 1

like 119901(sdot) or 119897(sdot) which accept only one symbol as an inputare unary whereas binary predicates like inc(sdot sdot) and eq(sdot sdot)accept pairs of symbols as an input In a fuzzy predicate logicpredicates are interpreted by fuzzy relations instead of crisprelations For example a binary fuzzy relation eq is a functioneq119873times119873 rarr [0 1] assigning a real number 120582 isin [0 1] to everypair of objects from N In other words every two objects of119873 are equal to some degree The degree of equality of twoobjects a and b may be 1 or 0 as in the crisp case but may aswell be 09 expressing that a and b are almost equal In [1ndash4]the fuzzification of 119901(sdot) 119897(sdot) inc(sdot) and eq(sdot) predicates wasproposed

Similarly to [1ndash4] we define a bounded subset Dom sube 1198772

as the domain for our geometric exercises Predicates aredefined for two-dimensional subsets 119860 119861 119862 of Dom andassume values in [0 1] We may assume two-dimensionalsubsets and ignore subsets of lower dimension because everymeasurement and every digitization introduces a minimumamount of location uncertainty in the data [2] For the point-predicate 119901(sdot) the result of Cartesian geometry involves aCartesian point that does not change when the point isrotated rotation-invariance seems to be amain characteristicof ldquopoint likenessrdquo with respect to geometric operations itshould be kept when defining a fuzzy predicate expressing theldquopoint likenessrdquo of extended subsets of 1198772 As a preliminarydefinition let

120579min (119860)

= min119905

10038161003816100381610038161003816ch (119860) cap 119888 (119860) + 119905 sdot 119877120572 sdot (0 1)

119879| 119905 isin R

10038161003816100381610038161003816

120579max (119860)

= max119905

10038161003816100381610038161003816ch (119860) cap 119888 (119860) + 119905 sdot 119877120572 sdot (0 1)

119879| 119905 isin R

10038161003816100381610038161003816

(4)

be the minimal and maximal diameter of the convex hullch(119860) of 119860 sube Dom respectively The convex hull regularizesthe setsA andB and eliminates irregularities 119888(119860)denotes thecentroid of ch(119860) and119877

120572denotes the rotationmatrix by angle

120572 (Figure 1(a)) [1ndash4]Since A is bounded ch(119860) and 119888(119860) exist We can now

define the fuzzy point-predicate 119901(sdot) by

119901 (119860) =

120579min (119860)

120579max (119860) (5)

119860 sube Dom119901(sdot) expresses the degree to which the convexhull of a Cartesian point set A is rotation-invariant if 119901(119860) =1 then ch(119860) is perfectly rotation-invariant it is a disc Here120579max(119860) = 0 always holds because A is assumed to be two-dimensional Converse to 119901(sdot) the fuzzy line-predicate

119897 (119860) = 1 minus 119901 (119860) (6)

Let us express the degree to which a Cartesian point set119860 sube Dom is sensitive to rotation Since we only regard convexhulls 119897(sdot) disregards the detailed shape and structure of A butonly measures the degree to which A is directed

A fuzzy version of the incidence-predicate inc(sdot sdot) is abinary fuzzy relation between Cartesian point sets 119861 sube Dom

inc (119860 119861)

= max(|ch (119860) cap ch (119861)||ch (119860)|

|ch (119860) cap ch (119861)||ch (119861)|

)

(7)

measures the relative overlaps of the convex hulls of A andB and selects the greater one Here |ch(119860)| denotes the areaoccupied by ch(119860)The greater inc(119860 119861) ldquothemore incidentrdquoA and B if 119860 sube 119861 or 119861 sube 119860 then inc(119860 119861) = 1 and A and Bare considered incident to degree one

Conversely to inc(sdot sdot) a graduated equality predicateeq(sdot sdot) between the boundedCartesian point sets119860 119861 sube Domcan be defined as follows

eq (119860 119861)

= min(|ch (119860) cap ch (119861)||ch (119860)|

|ch (119860) cap ch (119861)||ch (119861)|

)

(8)


where eq(119860 119861) measures the minimal relative overlap of AandB whereasnoteq(119860 119861) = 1minuseq(119860 119861)measures the degreesto which the two point sets do not overlap if eq(119860 119861) asymp 0then A and B are ldquoalmost disjointrdquo

The following measure of ldquodistinctness of pointsrdquo dp(sdot)of two extended objects tries to capture this fact (Figure 1(b))We define

dp (119860 119861) = max(0 1 minusmax (120579max (119860) 120579max (119861))

120579max (ch (119860 cup 119861))) (9)

where dp(119860 119861) expresses the degree to which ch(119860) andch(119861) are distinct the greater dp(119860 119861) the more A andB behave like distinct Cartesian points with respect toconnection Indeed for Cartesian points a and b we wouldhave dp(119860 119861) = 1 If the distance between theCartesian pointsets A and B is infinitely big then dp(119860 119861) = 1 as well Ifmax(120579max(119860) 120579max(119861)) gt 120579max(ch(119860cup119861)) then dp(119860 119861) = 0

33 Formalization of Fuzzy Predicates To formalize fuzzypredicates defined in Section 32 both implication rarr andconjunction operators are defined as in Table 1

119860 and 119861 =

119886 + 119887

2

119886 + 119887 gt 1

0 119886 + 119887 le 1

(10)

119860 997888rarr 119861 =

1 minus 119886 + 119887

2

119886 gt 119887

1 119886 le 119887

(11)

In our further discussions we will also use the disjunctionoperator from the same table

119860 or 119861 =

119886 + 119887

2

119886 + 119887 lt 1

1 119886 + 119887 ge 1

(12)

Now let us redefine the set of fuzzy predicates (7)ndash(9)using proposed fuzzy logicrsquos operators

Proposition 1 If fuzzy predicate 119894119899119888(sdot sdot) is defined as in (7)and conjunction operator is defined as in (10) then

inc (119860 119861) =

119886 + 119887

2119886

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(13)

Proof Let us present (7) as follows

inc (119860 119861) = |119860 cap 119861|

min (|119860| |119861|) (14)

And given that

min (|119860| |119861|) = 119886 + 119887 minus |119886 minus 119887|2

(15)

from (7) and (10) we are getting (13)

It is important to notice that for the case when 119886 + 119887 gt 1in (13) the value of inc(119860 119861) ge 1 which means that (13) infact reduced into the following

inc (119860 119861)

=

1 119886 + 119887 gt 1 119886 = 119887 119886 gt 05 119887 gt 05

0 119886 + 119887 le 1

(16)

Proposition 2 If fuzzy predicate 119890119902(sdot sdot) is defined as in (8) anddisjunction operator is defined as in (12) then

eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(17)

Proof Let us rewrite (8) in the following way

eq (119860 119861) = min(119860 cap 119861119860

119860 cap 119861

119861

) (18)

Let us define 119875 = 119860 cap 119861119860 and 119876 = 119860 cap 119861119861 and given(10) we have got the following

119875 =

119886 + 119887

2119886

119886 + 119887 gt 1

0 119886 + 119887 le 1

119876 =

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(19)

Therefore given (15) we have the followingLet us use (19) in the expression of min in (15) and first

find the following

119875 + 119876 =

119886 + 119887

2119886

+

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

=

(119886 + 119887)2

2119886119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(20)

In the meantime we can show that the following is alsotaking place

119875 minus 119876 =

119886 + 119887

2119886

minus

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

=

1198872minus 1198862

2119886119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(21)


From (21) we are getting

|119875 minus 119876| =

1198872minus 1198862

2119886119887

119886 + 119887 gt 1 119887 gt 119886

1198862minus 1198872

2119886119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(22)

But from (18) we have the following

eq (119860 119861) = min (119875 119876) = 119875 + 119876 minus |119875 minus 119876|2

=

(119886 + 119887)2minus 1198872+ 1198862

2119886119887

119886 + 119887 gt 1 119887 gt 119886

(119886 + 119887)2minus 1198862+ 1198872

4119886119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

=

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(23)

Corollary 3 If fuzzy predicate 119890119902(119860 119861) is defined as (23)then the following type of transitivity is taking place

eq (119860 119862) 997888rarr eq (119860 119861) and eq (119861 119862) (24)

where 119860 119861 119862 sube Dom andDom is partially ordered space thatis either 119860 sube 119861 sube 119862 or vice versa (note both conjunction andimplication operations are defined in Table 1)

Proof From (17) we have

eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

eq (119861 119862) =

119887 + 119888

2119888

119887 + 119888 gt 1 119887 lt 119888

119887 + 119888

2119887

119887 + 119888 gt 1 119887 gt 119888

0 119887 + 119888 le 1

(25)

then

eq (119860 119861) and eq (119861 119862)

=

eq (119860 119861) + eq (119861 119862)2

eq (119860 119861) + eq (119861 119862) gt 1

0 eq (119860 119861) + eq (119861 119862) le 1

(26)

Meanwhile from (17) we have the following

eq (119860 119862) =

119886 + 119888

2119888

119886 + 119888 gt 1 119886 lt 119888

119886 + 119888

2119886

119886 + 119888 gt 1 119886 gt 119888

0 119886 + 119888 le 1

(27)

Case 1 (119886 lt 119887 lt 119888) From (26) we have

(eq (119860 119861) and eq (119861 119862))2

=

119886 + 119887

2119887

+

119887 + 119888

2119888

=

119886119888 + 2119887119888 + 1198872

4119887119888

(28)

From (27) and (28) we have to prove that

119886 + 119888

2119888

997888rarr

119886119888 + 2119887119888 + 1198872

4119887119888

(29)

But (29) is the same as (2119886119887 + 2119887119888)4119887119888 rarr (119886119888 + 2119887119888 +

1198872)4119887119888 from which we get 2119886119887 rarr 119886119888 + 119887

2

Fromdefinition of implication in fuzzy logic (11) and sincefor 119886 lt 119887 lt 119888 condition 2119886119887 lt 119886119888+1198872 is taking place therefore2119886119887 rarr 119886119888 + 119887

2= 1

Case 2 (119886 gt 119887 gt 119888) From (26) we have

(eq (119860 119861) and eq (119861 119862))2

=

119886 + 119887

2119886

+

119887 + 119888

2119887

=

119886119888 + 2119886119887 + 1198872

4119886119887

(30)


119886 + 119888

2119886

997888rarr

119886119888 + 2119886119887 + 1198872

4119886119887

(31)



2

Fromdefinition of implication in fuzzy logic (11) and sincefor 119886 gt 119887 gt 119888 condition 2119887119888 lt 119886119888+1198872 is taking place therefore2119887119888 rarr 119886119888 + 119887

2= 1

Proposition 4 If fuzzy predicate 119889119901(sdot sdot) is defined as in (9)and disjunction operator is defined as in (12) then

dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(32)


Proof From (9) we get the following

dp (119860 119861) = max 0 1 minus max (119860 119861)119860 cup 119861

(33)

Given that max(119860 119861) = (119886 + 119887 + |119886 minus 119887|)2 from (33) and (9)we are getting the following

dp (119860 119861)

=

max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

119886 + 119887 lt 1

max0 1 minus 119886 + 119887 + |119886 minus 119887|2

119886 + 119887 ge 1

(34)

(1) From (34) we have

max0 1 minus 119886 + 119887 + |119886 minus 119887|2

= max0 2 minus 119886 minus 119887 minus |119886 minus 119887|2

=

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

(35)

(2) Also from (34) we have

max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

= max0 minus |119886 minus 119887|119886 + 119887

= 0 119886 + 119887 lt 1

(36)

From both (35) and (36) we have gotten that

dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(37)

34 Fuzzy Axiomatization of Incidence Geometry Using thefuzzy predicates formalized in Section 33 we propose theset of axioms as fuzzy version of incidence geometry in thelanguage of a fuzzy logic [5] as follows

(11986811015840) dp(119886 119887) rarr sup

119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]

(11986821015840) dp(119886 119887) rarr [119897(119888) rarr [inc(119886 119888) rarr [inc(119887 119888) rarr 119897(119888

1015840) rarr

[inc(119886 1198881015840) rarr [inc(119887 1198881015840) rarr eq(119888 1198881015840)]]]]](11986831015840) 119897(119888) rarr sup

119886119887119901(119886) and 119901(119887) and noteq(119886 119887) and inc(119886 119888) and

inc(119887 119888)(11986841015840) sup119886119887119888119889

[119901(119886) and 119901(119887) and 119901(119888) and 119897(119889) rarr not(inc(119886 119889) andinc(119887 119889) and inc(119888 119889))]

In axioms (11986811015840)ndash(11986841015840) we also use a set of operations (10)ndash(12)

Proposition 5 If fuzzy predicates 119889119901(sdot sdot) and 119894119899119888(sdot sdot) aredefined like (37) and (13) respectively then axiom (1198681

1015840) is ful-

filled for the set of logical operators from a fuzzy logic [5] (Forevery two distinct points a and b at least one line l exists ieincident with a and b)

Proof From (16)

inc (119860 119862)

=

1 119886 + 119888 gt 1 119886 = 119888 119886 gt 05 119888 gt 05

0 119886 + 119888 le 1

inc (119861 119862)

=

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1

inc (119860 119862) and inc (119861 119862) = inc (119860 119862) + inc (119861 119862)2

equiv 1

(38)

sup119888[119897(119888) and inc(119886 119888) and inc(119887 119888)] and given (38) sup

119888[119897(119888) and 1] =

0 and 1 equiv 05 From (37) and (10) dp(119886 119887) le 05 and we aregetting dp(119886 119887) le sup

119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]

Proposition 6 If fuzzy predicates 119889119901(sdot sdot) 119890119902(sdot sdot) and 119894119899119888(sdot sdot)are defined like (37) (17) and (16) respectively then axiom(11986821015840) is fulfilled for the set of logical operators from a fuzzy logic

[5] (for every two distinct points a and b at least one line lexists ie incident with a and b and such a line is unique)

Proof Let us take a look at the following implication

inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) (39)

But from (27) we have

eq (119862 1198621015840) =

119888 + 1198881015840

21198881015840 119888 + 119888

1015840gt 1 119888 lt 119888

1015840

119888 + 1198881015840

2119888

119888 + 1198881015840gt 1 119888 gt 119888

1015840

0 119888 + 1198881015840le 1

(40)

And from (16)

inc (119861 119862) =

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1

(41)

From (40) and (41) we see that inc(119861 119862) le eq(119862 1198621015840) whichmeans that

inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) equiv 1 (42)

therefore the following is also true

inc (119886 1198881015840) 997888rarr [inc (119887 1198881015840) 997888rarr eq (119888 1198881015840)] equiv 1 (43)

Now let us take a look at the following implication inc(119887119888) rarr 119897(119888

1015840) Since inc(119887 119888) ge 119897(1198881015840) we are getting inc(119887 119888) rarr

119897(1198881015840) equiv 0 Taking into account (43) we have the following


inc (119887 119888) 997888rarr 119897 (1198881015840) 997888rarr [inc (119886 1198881015840) 997888rarr [inc (119887 1198881015840) 997888rarr eq (119888 1198881015840)]] equiv 1 (44)

Since from (16) inc(119886 119888) le 1 then with taking into account(44) we have gotten the following

inc (119886 119888) 997888rarr [inc (119887 119888) 997888rarr 119897 (1198881015840) 997888rarr [inc (119886 1198881015840) 997888rarr [inc (119887 1198881015840) 997888rarr eq (119888 1198881015840)]]] equiv 1 (45)

Since 119897(119888) le 1 from (45) we are getting

119897 (119888) 997888rarr [inc (119886 119888) 997888rarr [inc (119887 119888) 997888rarr 119897 (1198881015840) 997888rarr [inc (119886 1198881015840) 997888rarr [inc (119887 1198881015840) 997888rarr eq (119888 1198881015840)]]]] equiv 1 (46)

Finally because dp(119886 119887) le 1 we have

dp (119886 119887) le 119897 (119888) 997888rarr [inc (119886 119888) 997888rarr [inc (119887 119888) 997888rarr 119897 (1198881015840) 997888rarr [inc (119886 1198881015840) 997888rarr [inc (119887 1198881015840) 997888rarr eq (119888 1198881015840)]]]] (47)


1015840) is

fulfilled for the set of logical operators from a fuzzy logic [5](Every line is incident with at least two points)

Proof It was already shown in (38) that


equiv 1 (48)

And from (17) we have

eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(49)

The negation noteq(119860 119861) will be

noteq (119860 119861) =

119887 minus 119886

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 minus 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(50)

Given (38) and (50) we get

noteq (119860 119861) and 1

=

[1 + (119887 minus 119886) 2119887]

2

119886 + 119887 gt 1 119886 lt 119887

[1 + (119886 minus 119887) 2119886]

2

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

=

3119887 minus 119886

4119887

119886 + 119887 gt 1 119886 lt 119887

3119886 minus 119887

4119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(51)

since noteq(119860 119861) and 1 equiv 05 | 119886 = 1 119887 = 1 from which we aregetting sup

119886119887119901(119886) and 119901(119887) andnoteq(119886 119887) and inc(119886 119888) and inc(119887 119888) =

1 and 05 = 075

And given that 119897(119888) le 075 we are getting 119897(119888) rarr

sup119886119887119901(119886) and 119901(119887) and noteq(119886 119887) and inc(119886 119888) and inc(119887 119888) equiv 1

Proposition 8 If fuzzy predicate 119894119899119888(sdot sdot) is defined like (16)then axiom (11986841015840) is fulfilled for the set of logical operators froma fuzzy logic [5] (At least three points exist that are not incidentwith the same line)


L

M

Q

P

Figure 2 Two extended points do not uniquely determine thelocation of an incident extended line


inc (119860119863)

=

1 119886 + 119889 gt 1 119886 = 119889 119886 gt 05 119889 gt 05

0 119886 + 119889 le 1

inc (119861 119863)

=

1 119887 + 119889 gt 1 119887 = 119889 119887 gt 05 119889 gt 05

0 119887 + 119889 le 1

inc (119862119863)

=

1 119888 + 119889 gt 1 119888 = 119889 119888 gt 05 119889 gt 05

0 119888 + 119889 le 1

(52)


inc (119860119863) and inc (119861 119863) = inc (119860119863) + inc (119861 119863)2

equiv 1 (53)

and (inc(119886 119889) and inc(119887 119889) and inc(119888 119889)) = 1 and inc(119888 119889) equiv 1where we have not(inc(119886 119889) and inc(119887 119889) and inc(119888 119889)) equiv 0 Since119897(119889) equiv 0 | 119889 = 1 we are getting 119897(119889) = not(inc(119886 119889) andinc(119887 119889) and inc(119888 119889)) which could be interpreted like 119897(119889) rarrnot(inc(119886 119889) and inc(119887 119889) and inc(119888 119889)) = 1 from which we finallyget sup

119886119887119888119889[119901(119886) and 119901(119887) and 119901(119888) and 1] equiv 1

35 Equality of Extended Lines Is Graduated In [10] it wasshown that the location of the extended points creates aconstraint on the location of an incident extended line It wasalsomentioned that in traditional geometry this location con-straint fixes the position of the line uniquely It is not true incase of extended points and lines Consequently Euclidrsquos firstpostulate does not apply Figure 2 shows that if two distinctextended points P and Q are incident (ie overlap) with twoextended lines L and M then L and M are not necessarilyequal

Yet in most cases L and M are ldquocloser togetherrdquo that isldquomore equalrdquo than arbitrary extended lines that have only oneor no extended point in commonThe further P and Qmoveapart from each other the more similar L and M becomeOne way to model this fact is to allow degrees of equality forextended lines In other words the equality relation is gradu-ated it allows for not only Boolean values but also values inthe whole interval [0 1]

36 Incidence of Extended Points and Lines As it was demon-strated in [10] there is a reasonable assumption to classify

an extended point and an extended line as incident if theirextended representations in the underlying metric spaceoverlap We do this by modelling incidence by the subsetrelation

Definition 9 For an extended point P and an extended line Lwe define the incidence relation by

119877inc (119875 119871) fl (119875 sube 119871) isin 0 1 (54)where the subset relation sube refers to P and L as subsets of theunderlying metric space

The extended incidence relation (54) is a Boolean relationassuming either the truth value 1 (true) or the truth value 0(false) It is well known that since a Boolean relation is a spe-cial case of a graduated relation that is since 0 1 sub [0 1]we will be able to use relation (54) as part of fuzzified Euclidrsquosfirst postulate later on

37 Equality of Extended Points and Lines As stated inprevious sections equality of extended points and equality ofextended lines are a matter of degree Geometric reasoningwith extended points and extended lines relies heavily onthe metric structure of the underlying coordinate spaceConsequently it is reasonable to model graduated equality asinverse to distance

371 Metric Distance In [10] it was mentioned that a pseudometric distance or pseudo metric is a map 119889 1198722 rarr R+

from domain M into the positive real numbers (includingzero) which is minimal and symmetric and satisfies thetriangle inequality

forall119901 119902 isin [0 1]

dArr

119889 (119901 119901) = 0

119889 (119901 119902) = 119889 (119902 119901)

119889 (119901 119902) + 119889 (119902 119903) ge 119889 (119901 119903)

(55)

d is called a metric if the following takes place

119889 (119901 119902) = 0 lArrrArr

119901 = 119902

(56)

Well-known examples of metric distances are the Euclideandistance or the Manhattan distance The ldquoupside-down-versionrdquo of a pseudo metric distance is a fuzzy equivalencerelation with respect to a proposed t-norm The next sectionintroduces the logical connectives in a proposed t-normfuzzy logicWewill use this particular fuzzy logic to formalizeEuclidrsquos first postulate for extended primitives in Section 5The reason for choosing a proposed fuzzy logic is its strongconnection to metric distance

372 The t-Norm

Proposition 10 In proposed fuzzy logic the operation of con-junction (10) is a t-norm


Proof The function 119891(119901 119902) is a t-norm if the following hold(1) Commutativity 119901 and 119902 = 119902 and 119901(2) Associativity (119901 and 119902) and 119903 = 119901 and (119902 and 119903)(3) Monotonicity 119901 le 119902 119901 and 119903 le 119902 and 119903(4) Neutrality 1 and 119901 = 119901(5) Absorption 0 and 119901 = 0

Commutativity Consider

119891 (119901 119902) = 119875 cap 119876 =

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

119891 (119902 119901) = 119876 cap 119875 =

119902 + 119901

2

119902 + 119901 gt 1

0 119902 + 119901 le 1

(57)

therefore 119891(119901 119902) = 119891(119902 119901)

Associativity

Case 1 (119891(119901 119902) and 119903) Consider

119891 (119901 119902) =

119901 + 119902

2

119901 + 119902 gt 1 997904rArr

119891 (119901 119902) and 119903 =

119891 (119901 119902) + 119903

2

119891 (119901 119902) + 119903 gt 1

0 119891 (119901 119902) + 119903 le 1

=

119901 + 119902 + 2119903

4

119901 + 119902

2

+ 119903 gt 1

0

119901 + 119902

2

+ 119903 le 1

(58)

from where we have that

1198911(119901 119903) =

119901 + 119902 + 2119903

4

119901 + 119902 + 2119903 gt 2

0 119901 + 119902 + 2119903 le 2

(59)

In other words 1198911(119901 119903) sube (05 1] | 119901 + 119902 + 2119903 gt 2 and

1198911(119901 119903) = 0 | 119901 + 119902 + 2119903 le 2For the case 119901 and 119891(119902 119903) we are getting results similar to

(59)

1198912(119901 119903) =

119902 + 119903 + 2119901

4

119902 + 119903 + 2119901 gt 2

0 119902 + 119903 + 2119901 le 2

(60)

that is 1198912(119901 119903) sube (05 1] | 119902 + 119903 + 2119901 gt 2 and 119891

2(119901 119903) = 0 |

119902 + 119903 + 2119901 le 2 Consider 1198911(119901 119903) asymp 119891

2(119901 119903)

Monotonicity If 119901 le 119902 rArr 119901 and 119903 le 119902 and 119903 then given

119901 and 119903 =

119901 + 119903

2

119901 + 119903 gt 1

0 119901 + 119903 le 1

119902 and 119903 =

119902 + 119903

2

119902 + 119903 gt 1

0 119902 + 119903 le 1

(61)

we are getting (119901 + 119903)2 le (119902 + 119903)2 rArr 119901 + 119903 le 119902 + 119903 rArr 119901 le

119902 | 119901 + 119903 gt 1 and 119902 + 119903 gt 1 whereas for the case 119901 + 119903 le 1119902 + 119903 le 1 rArr 0 equiv 0

Neutrality Consider

1 and 119901 =

1 + 119901

2

1 + 119901 gt 1

0 1 + 119901 le 1

=

1 + 119901

2

119901 gt 0

0 119901 le 0

=

1 + 119901

2

119901 gt 0

0 119901 = 0

(62)

from which the following is apparent

1 and 119901 =

1 + 119901

2

119901 isin (0 1)

119901 119901 = 0 119901 = 1

(63)

Absorption Consider

0 and 119901 =

119901

2

119901 gt 1

0 119901 le 1

(64)

since 119901 isin [0 1] rArr 0 and 119901 equiv 0

373 Fuzzy Equivalence Relations As mentioned above theldquoupside-down-versionrdquo of a pseudometric distance is a fuzzyequivalence relation with respect to the proposed t-norm andA fuzzy equivalence relation is a fuzzy relation 119890 1198722 rarr[0 1] on a domain M which is reflexive symmetric and and-transitive

forall119901 119902 isin [0 1]

dArr

119890 (119901 119901) = 1

119890 (119901 119902) = 119890 (119902 119901)

119890 (119901 119902) and 119890 (119902 119903) le 119890 (119901 119903)

(65)

Proposition 11 If fuzzy equivalence relation is defined(Table 1) as

119890 (119901 119902) = 119875 larrrarr

119876 =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(66)

then conditions (65) are satisfied

Proof

(1) Reflexivity 119890(119901 119901) = 1 comes from (66) because 119901 equiv 119901


(2) Symmetricity 119890(119901 119902) = 119890(119902 119901) Consider the following

119890 (119901 119902) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(67)

but

119890 (119902 119901) =

1 minus 119902 + 119901

2

119902 gt 119901

1 119902 = 119901

1 minus 119901 + 119902

2

119902 lt 119901

(68)

therefore 119890(119901 119902) equiv 119890(119902 119901)

(3) Transitivity 119890(119901 119902) and 119890(119902 119903) le 119890(119901 119903) | forall119901 119902 119903 isin 119871[0 1]-lattice

From (66) let

1198651(119901 119903) = 119890 (119901 119903) =

1 minus 119901 + 119903

2

119901 gt 119903

1 119901 = 119903

1 minus 119903 + 119901

2

119901 lt 119903

(69)

119890 (119902 119903) =

1 minus 119902 + 119903

2

119902 gt 119903

1 119902 = 119903

1 minus 119903 + 119902

2

119902 lt 119903

(70)

then

1198652(119901 119903) = 119890 (119901 119902) and 119890 (119902 119903)

=

119890 (119901 119902) + 119890 (119902 119903)

2

119890 (119901 119902) + 119890 (119902 119903) gt 1

0 119890 (119901 119902) + 119890 (119902 119903) le 1

(71)

But

1198652(119901 119903) =

119890 (119901 119902) + 119890 (119902 119903)

2

=

((1 minus 119901 + 119902) 2 + (1 minus 119902 + 119903) 2)

2

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

((1 minus 119902 + 119901) 2 + (1 minus 119903 + 119902) 2)

2

119901 lt 119902 lt 119903

=

2 minus 119901 + 119903

4

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

2 minus 119903 + 119901

4

119901 lt 119903 lt 119903

(72)

Now compare (72) and (69) It is apparent that 119903 gt 119901 rArr (2 minus

119901+119903)4 lt (1minus119901+119903)2 hArr 119903minus119901 lt 2(119903minus119901)The same is true for119901 gt 119903 rArr (2minus119903+119901)4 lt (1minus119903+119901)2 hArr 119901minus119903 lt 2(119901minus119903) Andlastly (119890(119901 119902) + 119890(119902 119903))2 equiv 119890(119901 119903) equiv 1 when 119901 = 119903 Giventhat 119865

2(119901 119903) = 119890(119901 119902) and 119890(119902 119903) equiv 0 119890(119901 119902) + 119890(119902 119903) le 1 we

are getting the proof of the fact that 1198652(119901 119903) le 119865

1(119901 119903) hArr

119890(119901 119902) and 119890(119902 119903) le 119890(119901 119903) | forall119901 119902 119903 isin 119871[0 1]

Note that relation 119890(119901 119902) is called a fuzzy equality relationif additionally separability holds 119890(119901 119902) = 1 hArr 119901 = 119902Let us define a pseudo metric distance 119889(119901 119902) for domain Mnormalized to 1 as

119890 (119901 119902) = 1 minus 119889 (119901 119902) (73)


119889 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

=

1 +1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

119901 = 119902

0 119901 = 119902

(74)

374 Approximate Fuzzy Equivalence Relations In [11] it wasmentioned that graduated equality of extended lines compelsgraduated equality of extended points Figure 3(a) sketches asituation where two extended lines L and M intersect in anextended point P If a third extended line 1198711015840 is very similar toL its intersection with M yields an extended point 1198751015840 whichis very similar to P It is desirable to model this fact To do soit is necessary to allow graduated equality of extended points

Figure 3(b) illustrates that an equality relation betweenextended objects need not be transitive This phenomenon iscommonly referred to as the Poincare paradox The Poincareparadox is named after the famous French mathematicianand theoretical physicist Poincare who repeatedly pointedthis fact out for example in [12] referring to indiscernibilityin sensations andmeasurements Note that this phenomenonis usually insignificant if positional uncertainty is caused bystochastic variability In measurements the stochastic vari-ability caused by measurement inaccuracy is usually muchgreater than the indiscernibility caused by limited resolutionFor extended objects this relation is reversed the extension ofan object can be interpreted as indiscernibility of its con-tributing points In the present paper we assume that theextension of an object is being compared with the indetermi-nacy of its boundaryGerla shows thatmodelling the Poincareparadox in graduated context transitivity may be replaced bya weaker form [13]

119890 (119901 119902) and 119890 (119902 119903) and dis (119902) le 119890 (119901 119903) (75)

Here dis 119872 rarr [0 1] is a lower-bound measure(discernibility measure) for the degree of transitivity that ispermitted by q A pair (119890 dis) that is reflexive symmetric


L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)

Figure 3 (a) Graduated equality of extended lines compels graduated equality of extended points (b) Equality of extended lines is nottransitive

and weakly transitive (75) is called an approximate fuzzy and-equivalence relation Let us rewrite (75) as follows

1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865

1(119901 119903) are defined in (72) and (69) corre-

spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)

From (77) in order to satisfy condition (76) we have

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

dis (119902) gt 1 minus2 minus 119903 + 119901

4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)

By using (79) in (77) we are getting that forall119901 119902 119903 isin [0 1] rArr1198652(119901 119903) and dis(119902) equiv 05 From (69) we are getting forall119901 119903 isin

[0 1] rArr 1198651(119901 119903) isin [05 1] and subsequently inequality (76)

holds

In [11] it was also mentioned that an approximate fuzzy and-equivalence relation is the upside-down-version of a so-calledpointless pseudo metric space (120575 119904)

120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)

Here 120575 119872 rarr R+ is a (not necessarily metric) distancebetween extended regions and 119904 119872 rarr R+ is a size measureand we are using an operation disjunction (12) also shown inTable 1 Inequality 120575(119902 119903) or 119904(119902) ge 120575(119901 119903) is a weak form ofthe triangle inequality It corresponds to the weak transitivity(75) of the approximate fuzzy and-equivalence relation e In casethe size of the domainM is normalized to 1 e and dis can berepresented by [13]

119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)

Proposition 12 If a distance between extended regions 120575(119901 119902)from (80) and pseudo metric distance 119889(119901 119902) for domain Mnormalized to 1 are the same that is 120575(119901 119902) = 119889(119901 119902) theninequality 120575(119901 119902) or 120575(119902 119903) or 119904(119902) ge 120575(119901 119903) holds


120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)

From (83) and (84) the following is apparent

120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)

Now we have to show that size measure 119904(119902) gt 0 From (79)we have

119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)

It is apparent that 119904(119902) isin (025 1] | forall119903 119901 119902 isin [0 1] thereforefrom (84) (85) and (86) 120575(119901 119902) or 120575(119902 119903) or 119904(119902) ge 120575(119901 119903)

holds

Noting 120575(119901 119903) from (84) we have forall119903 119901 isin [0 1] rArr

120575(119901 119903) = (1 + |119901 minus 119903|)2 isin [0 1] But as it was mentioned in[10] given a pointless pseudo metric space (120575 119904) for extendedregions on a normalized domain (81) define an approximatefuzzy and-equivalence relation (119890 dis) by simple logical nega-tion The so-defined equivalence relation on the one handcomplies with the Poincare paradox and on the other handit retains enough information to link two extended points (orlines) via a third For used fuzzy logic an example of a pointless

pseudo metric space is the set of extended points with thefollowing measures

120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)

It is easy to show that (86) and (87) are satisfied because from(74) 119889(119901 119902) isin [0 1] | forall119903 119901 119902 isin [0 1] A pointless metricdistance of extended lines can be defined in the dual space[10]

120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)

375 Boundary Conditions for Granularity As it was men-tioned in [10] in exact coordinate geometry points and linesdo not have size As a consequence distance of points doesnot matter in the formulation of Euclidrsquos first postulate Ifpoints and lines are allowed to have extension both size anddistancematter Figure 4 depicts the location constraint on anextended lineL that is incidentwith the extendedpointsP andQ

The location constraint can be interpreted as tolerance inthe position of L In Figure 4(a) the distance of P and Q islarge with respect to the sizes of P and Q and with respectto the width of L The resulting positional tolerance for L issmall In Figure 4(b) the distance of P and Q is smaller thanthat in Figure 4(a) As a consequence the positional tolerancefor L becomes larger In Figure 4(c) P and Q have thesame distance as in Figure 4(a) but their sizes are increasedAgain positional tolerance of L increases As a consequence


L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)

Figure 4 Size and distance matter

L

L998400

QP

Figure 5 P and Q are indiscernible for L

formalization of Euclidrsquos first postulate for extended primi-tivesmust take all three parameters into account the distanceof the extended points their size and the size of the incidentline

Figure 5 illustrates this case despite the fact that P and Qare distinct extendedpoints that are both incidentwithL theydo not specify any directional constraint for L Consequentlythe directional parameter of the extended lines L and 1198711015840 inFigure 5 may assume its maximum (at 90∘) If we measuresimilarity (ie graduated equality) as inverse to distance andif we establish a distancemeasure between extended lines thatdepends on all parameters of the lines parameter space thenL and 1198711015840 in Figure 5 must have maximum distance In otherwords their degree of equality is zero even though they aredistinct and incident with P and Q

The above observation can be interpreted as granularityif we interpret the extended line L in Figure 5 as a sensor thenthe extended points P andQ are indiscernible for L Note thatin this context grain size is not constant but depends on theline that serves as a sensor

Based on the above-mentioned information granularityenters Euclidrsquos first postulate if points and lines have exten-sion if two extended points P and Q are too close and theextended line L is too broad then P and Q are indiscernible

for L Since this relation of indiscernibility (equality) dependsnot only on P and Q but also on the extended line L whichacts as a sensor we denote it by 119890(119875 119876)[119871] where L serves asan additional parameter for the equality of P and Q

In [10] the following three boundary conditions to specifya reasonable behavior of 119890(119875 119876)[119871] were proposed

(1) If 119904(119871) ge 120575(119875 119876)+119904(119875)+119904(119876) then P andQ impose nodirection constraint on L (cf Figure 5) that is P andQ are indiscernible for L to degree 1 119890(119875 119876)[119871] = 1

(2) If 119904(119871) lt 120575(119875 119876) + 119904(119875) + 119904(119876) then P and Q imposesome direction constraint on L but in general donot fix its location unambiguously Accordingly thedegree of indiscernibility of P andQ lies between zeroand one 0 lt 119890(119875 119876)[119871] lt 1

(3) If 119904(119871) lt 120575(119875 119876) + 119904(119875) + 119904(119876) and 119875 = 119901 119876 = 119902 and119871 = 119897 are crisp then 119904(119871) = 119904(119875) = 119904(119876) = 0 Con-sequently p and q determine the direction of l unam-biguously and all positional tolerance disappears Forthis case we demand 119890(119875 119876)[119871] = 0

In this paper we are proposing an alternative approach to theone from [10] to model granulated equality

Proposition 13 If fuzzy equivalence relation 119890(119875 119876) is definedin (66) and the width 119904(119871) of extended line L is defined in (90)then 119890(119875 119876)[119871] the degree of indiscernibility of P and Q couldbe calculated as follows

119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)

and it would satisfy a reasonable behavior defined in (1)ndash(3)Here and is conjunction operator from Table 1


Proof From (10) (91) and (66) we have

119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)

therefore we have the following

(1) If P andQ impose no direction constraint on Lwhichmeans that 119904(119871) = 1 and 120575(119875 119876) = 0 rArr 119890(119875 119876) = 1then 119890(119875 119876)[119871] = 1 (proof of point (1))

(2) If P andQ impose some direction constraint on L butin general do not fix its location unambiguously thenfrom (92) and (93) we are getting

119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)

isin (0 1) (proof of point (2))(3) If119875 = 119901 119876 = 119902 and 119871 = 119897 are crisp whichmeans that

values of p and q are either 0 or 1 and since (119871) = 0then 119890(119875 119876)[119871] = 0 (proof of point (3))

4 Fuzzification of Euclidrsquos First Postulate

41 Euclidrsquos First Postulate Formalization In previous sectionwe identified and formalized a number of new qualities thatenter into Euclidrsquos first postulate if extended geometric prim-itives are assumedWe are now in the position of formulatinga fuzzified version of Euclidrsquos first postulate To do this wefirst split the postulate which is given as

ldquotwo distinct points determine a line uniquelyrdquo

into two subsentences

ldquoGiven two distinct points there exists at least one linethat passes through themrdquoldquoIf more than one line passes through them then theyare equalrdquo

These subsentences can be formalized in Boolean predi-cate logic as follows

forall119901 119902 exist119897 [119877inc (119901 119897) and 119877inc (119902 119897)]

forall119901 119902 119897 119898 [not (119901 = 119902)] and [119877inc (119901 119897) and 119877inc (119902 119897)]

and [119877inc (119901119898) and 119877inc (119902119898)] 997888rarr (119897 = 119898)

(95)

A verbatim translation of (95) into the syntax of a fuzzy logicwe use yields

inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872

[not119890 (119875 119876)] and [119877inc (119875 119871) and 119877inc (119876 119871)]

and [119877inc (119875119872) and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(97)

where PQ denote extended points and LM denote extendedlines The translated existence property (96) can be adoptedas it is but the translated uniqueness property (97) must beadapted to include granulated equality of extended points

In contrast to the Boolean case the degree of equality oftwo given extended points is not constant but depends on theextended line that acts as a sensor Consequently the termnot119890(119875 119876) on the left-hand side of (97)must be replaced by twoterms not119890(119875 119876)[119871] and not119890(119875 119876)[119872 one for each line L andM respectively

inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)

We have to use weak transitivity of graduated equality Forthis reason the discernibility measure of extended connection119875119876 between extended points P and Q must be added into(98)

inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872] and dis (119875119876)]

and [119877inc (119875 119871) and 119877inc (119876 119871)] and [119877inc (119875119872)

and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)

But from (92) we get

not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)

By using (100) in (99) we get


not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=

4 minus 2 times 119890 (119875 119876) minus 119904 (119871) minus 119904 (119872)

4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)

Since from (92)wehave [119877inc(119875 119871)and119877inc(119876 119871)]and[119877inc(119875119872)and119877inc(119876119872)] equiv 1 then (99) could be rewritten as follows

inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)

It means that the ldquosamenessrdquo of extended lines 119890(119871119872)depends on [not119890(119875 119876)[119871] and not119890(119875 119876)[M] and dis(119875119876)] only andcould be calculated by (101) and (79) respectively

42 Fuzzy Logical Inference for Euclidrsquos First Postulate Con-trary to the approach proposed in [10] which required alot of calculations we suggest using the same fuzzy logicand correspondent logical inference to determine the valueof 119890(119871119872) For this purpose let us represent values of119864(119901 119902 119897 119898) = not119890(119875 119876)[119871] and not119890(119875 119876)[119872] from (101) and119863(119901 119902) = dis(119875119876) from (79) functions Note that all valuesof these functions are lying within certain intervals that is119864(119901 119902 119897 119898) isin [119864min 119864max] and 119863(119901 119902) isin [119863min 119863max]In our case 119864(119901 119902 119897 119898) isin [0 1] and 119863(119901 119902) isin [0 075]We denote by E the fuzzy set forming linguistic variablesdescribed by triplets of the form 119864 = ⟨119864

119894 119880 ⟩ 119864

119894isin

119879(119906) forall119894 isin [0Card119880] where 119879119894(119906) is extended term set

of the linguistic variable ldquodegree of indiscernibilityrdquo fromTable 3 and is normal fuzzy set represented by membershipfunction 120583

119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is

universe set and Card119880 is power set of the set U We will usethe mapping 120572 rarr 119880 |119906

119894= Ent [(Card119880 minus 1) times 119864

119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)

To determine the estimates of the membership functionin terms of singletons from (103) in the form 120583

119864(119906119894)119906119894| forall119894 isin

[0Card119880] we propose the following procedure

120583 (119906119894) = 1 minus

1

Card119880 minus 1

times1003816100381610038161003816119906119894minus Ent [(Card119880 minus 1) times 119864119894]

1003816100381610038161003816

forall119894 isin [0Card119880] forall119864119894 isin [0 1]

(104)

We also denote by D the fuzzy set forming linguisticvariables described by triplets of the form 119863 = ⟨119863

119895 119880 ⟩

119863119895isin 119879(119906) forall119895 isin [0Card119880] where 119879

119895(119906) is extended term

set of the linguistic variable ldquodiscernibility measurerdquo fromTable 3 and is normal fuzzy set represented bymembershipfunction 120583

119863 119880 rarr [0 1]

We will use the mapping 120573 rarr 119880 |V119895= Ent[(Card119880minus

1) times 119863119895]| forall119895 isin [0Card119880] where

= int

119880

120583119863 (119906)

119906

(105)

On the other hand to determine the estimates of themembership function in terms of singletons from (105) in theform 120583

119863(119906119895)119906119895| forall119895 isin [0Card119880] we propose the following

procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816

119906119895minus Ent [(Card119880 minus 1) times

119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)

Let us represent 119890(119871119872) as a fuzzy set forming linguisticvariables described by triplets of the form 119878 = ⟨119878

119896 119881 ⟩

119878119896isin 119879(V) forall119896 isin [0Card119881] where 119879


set of the linguistic variable ldquoextended lines samenessrdquo fromTable 3 is normal fuzzy set represented by membershipfunction 120583

119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is

universe set and Card119881 is power set of the set 119881 We will usethe mapping 120574 rarr 119881|V


119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)

Again to determine the estimates of the membershipfunction in terms of singletons from (107) in the form120583119878(119908119896)V119896| forall119896 isin [0Card119881] we propose the following

procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1

times1003816100381610038161003816V119896minus Ent [(Card119881 minus 1) times 119878119896]

1003816100381610038161003816


(108)

To get estimates of values of 119890(119871119872) or ldquoextended lines same-nessrdquo represented by fuzzy set from (107) given the values


Table 3

Value of variable 119906119894 V119895isin 119880 V

119896isin 119881

forall119894 119895 119896 isin [0 10]ldquoDegree of indiscernibilityrdquo ldquoDiscernibility measurerdquo ldquoExtended lines samenessrdquoLowest Highest Nothing in common 0Very low Almost highest Very far 1Low High Far 2Bit higher than low Pretty high Bit closer than far 3Almost average Bit higher than average Almost average distance 4Average Average Average 5Bit higher than average Almost average Bit closer than average 6Pretty high Bit higher than low Pretty close 7High Low Close 8Almost highest Very low Almost the same 9Highest Lowest The same 10

of 119864(119901 119902 119897 119898) or ldquodegree of indiscernibilityrdquo and 119863(119901 119902) orldquodiscernibility measurerdquo represented by fuzzy sets from(103) and from (105) respectively we will use a FuzzyConditional Inference Rule formulated by means of ldquocommonsenserdquo as the following conditional clause

119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)

THEN (E is Q) rdquo(109)

In words we use fuzzy conditional inference of the followingtype

Ant1 If 119904 is 1198751 and 119889 is 1198752 then 119890 is 119876Ant2 119904 is 11987511015840 and 119889 is 11987521015840

Cons 119890 is 1198761015840(110)

where 1198751 11987511015840 1198752 11987521015840 sube 119880 and 1198761198761015840 sube 119881

Now for fuzzy sets (103) (105) and (107) a binaryrelationship for the fuzzy conditional proposition of the typeof (109) and (110) for fuzzy logic we use so far is defined as

119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)

Given (11) expression (111) looks like

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well

known that given a unary relationship119877(1198601(119904 119889)) = 1198751

1015840cap11987521015840

one can obtain the consequence 119877(1198602(119890)) by applying

compositional rule of inference (CRI) to 119877(1198601(119904 119889)) and

119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)


But for practical purposes we will use another FuzzyConditional Rule (FCR)

119877 (1198601 (119904 119889) 1198602 (

119890))

= (119875 times 119881 997888rarr 119880 times 119876) cap (not119875 times 119881 997888rarr 119880 times not119876)

= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))

and ((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)

The FCR from (115) gives more reliable results

43 Example To build a binary relationship matrix of type(114) we use a conditional clause of type (109)

119875 = ldquoIF (119878 is ldquolowestrdquo) AND (119863 is ldquohighestrdquo)

THEN (119864 is ldquonothing in commonrdquo) rdquo(116)

Note that values ldquolowestrdquo of a linguistic variable ldquodegree ofindiscernibilityrdquo ldquohighestrdquo of a linguistic variable ldquodiscernibil-ity measurerdquo and ldquonothing in commonrdquo of a linguistic variableldquoextended lines samenessrdquo have come from Table 3 This par-ticular knowledge is based on common sense and representsa simple human perception of geometrical facts It is worthmentioning that (116) might be equivalently replaced by thefollowing knowledge statement

119875 = ldquoIF (119878 is ldquohighestrdquo)AND (119863 is ldquolowestrdquo)

THEN (119864 is ldquothe samerdquo) rdquo(117)

To build membership functions for fuzzy sets S D and E weuse (104) (106) and (108) respectively

In (116) the membership functions for fuzzy set S (forinstance) would look like

120583119904 (ldquolowestrdquo) = 1

0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)

which are the same membership functions we use for fuzzysets D and E

From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01

Suppose from (101) a current estimate of 119864(119901 119902 119897 119898) =06 and from (79) 119863(119901 119902) = 025 By using (104) and (106)respectively we got (see Table 3)

120583119864(ldquobit higher than averagerdquo)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10

120583119863(ldquopretty highrdquo)

=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)

By applying compositional rule of inference (CRI)to 119877(119860

1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860

2(119890) we got the

following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+

085 + 076 + 067 + 058 + 049 + 0310Since the maximum value of membership function of the

consequence (for two singletons 084 and 085) is given as

119877 (1198602(1198901015840)) = 08 (121)

then it is obvious that the value of fuzzy set S is lying betweenterms ldquoalmost average distancerdquo and ldquoaverage distancerdquo (seeTable 3) which means that

119890 (119871119872) isin [05 06] (122)


5 Conclusion

In [1ndash4] it was shown that straight forward interpretations ofthe connection of extended points do not satisfy the incidenceaxioms of Euclidean geometry in a strict senseWe formalizedthe axiom system of Boolean-Euclidean geometry by the lan-guage of the fuzzy logic [5] We also addressed fuzzificationof Euclidrsquos first postulate by using the same fuzzy logic Fuzzyequivalence relation ldquoextended lines samenessrdquo is introducedFor its approximation we use fuzzy conditional inferencewhich is based on proposed fuzzy ldquodegree of indiscernibilityrdquoand ldquodiscernibility measurerdquo of extended points

Competing Interests

The author declares that he has no competing interests

References

[1] G Wilke ldquoEquality in approximate tolerance geometryrdquo inProceedings of the 7th IEEE International Conference IntelligentSystems (IS rsquo14) pp 365ndash376 Springer International WarsawPoland September 2014

[2] G Wilke ldquoGranular geometryrdquo in Towards the Future of FuzzyLogic vol 325 pp 79ndash115 Springer 2015

[3] G Wilke ldquoApproximate geometric reasoning with extendedgeographic objectsrdquo in Proceedings of the ISPRS-COST Work-shop onQuality Scale and Analysis Aspects of CityModels LundSweden December 2009

[4] G Wilke Towards approximate tolerance geometry for gismdashaframework for formalizing sound geometric reasoning under posi-tional tolerance [PhD thesis] Vienna University of Technology2012

[5] R A Aliev and A Tserkovny ldquoSystemic approach to fuzzy logicformalization for approximate reasoningrdquo Information Sciencesvol 181 no 6 pp 1045ndash1059 2011

[6] R A Aliev and O H HuseynovDecisionTheory with ImperfectInformation World Scientific 2014

[7] R A Aliev and O H Huseynov ldquoFuzzy geometry-based deci-sion making with unprecisiated visual informationrdquo Interna-tional Journal of InformationTechnologyampDecisionMaking vol13 no 5 pp 1051ndash1073 2014

[8] R A Aliev ldquoDecision making on the basis of fuzzy geometryrdquoin Fundamentals of the Fuzzy Logic-Based GeneralizedTheory ofDecisions pp 217ndash230 Springer Berlin Germany 2013

[9] D HilbertGrundlagen der Geometrie Teubner StudienbuecherMathematik 1962

[10] G Wilke and A U Frank ldquoTolerance geometry Euclidrsquos firstpostulate for points and lines with extensionrdquo in Proceedingsof the 18th ACM SIGSPATIAL International Conference onAdvances in Geographic Information Systems (ACM-GIS rsquo10) pp162ndash171 San Jose Calif USA November 2010

[11] H Busemann and P J Kelly Projective Geometry and ProjectiveMetrics Academic Press 1953

[12] H Poincare Science and Hypothesis Walter Scott London UK1905

[13] G Gerla ldquoApproximate similarities and Poincare paradoxrdquoNotre Dame Journal of Formal Logic vol 49 no 2 pp 203ndash2262008

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Stochastic AnalysisInternational Journal of


for defining the notion of parallelism of two lines as a derivedconcept but does not permit expressing betweenness or con-gruency relations which are assumed primitives in Hilbertrsquossystem [9] The complete axiom set of Euclidean geometryprovides a greater number of construction operators thanincidence geometry Incidence geometry has very limitedexpressive power when compared with the full axiom system

The combined incidence axioms (I1) and (I2) state that itis always possible to connect two distinct points by a uniqueline In case of coordinate points a and b Cartesian geometryprovides a formula for constructing this unique line

119897 = 119886 + 119905 (119887 minus 119886) | 119905 isin 119877 (1)

As it was shown in [1ndash4] when we want to connect twoextended geographic objects in a similar way there is nocanonical way of doing so We cannot refer to an existingmodel like the Cartesian algebra Instead a new way of inter-preting geometric primitives must be found such that theinterpretation of the incidence relation respects the unique-ness property (I2)

Similarly to [1ndash4] we will refer to extended objects thatplay the geometric role of points and lines by extendedpoints and extended lines respectively Section 3 gives a briefintroduction on proposed fuzzy logic and discusses possibleinterpretations of fuzzy predicates for extended geometricprimitives The fuzzy logic from [5] is introduced as pos-sible formalism for approximate geometric reasoning withextended objects and based on extended geometric primitivesfuzzification of the incidence axioms (I1)ndash(I4) is investigated

3 Fuzzification of Incidence Geometry

31 Proposed Fuzzy Logic Let forall119901 119902 isin [0 1] and continuousfunction (119901 119902) = 119901 minus 119902 which defines a distance between 119901and 119902 Notice that 119865(119901 119902) isin [minus1 1] where 119865(119901 119902)min

= minus1

and 119865(119901 119902)max= 1 When normalized the value of 119865(119901 119902) is

defined as follows

119865 (119901 119902)norm

=

119865 (119901 119902) minus 119865 (119901 119902)min

119865 (119901 119902)max

minus 119865 (119901 119902)min

=

119865 (119901 119902) + 1

2

=

119901 minus 119902 + 1

2

(2)

It is clear that 119865(119901 119902)norm isin [0 1] This function repre-sents the value of ldquoclosenessrdquo between two values (potentiallyantecedent and consequent) defined within single intervalwhich therefore could play significant role in formulation ofan implication operator in a fuzzy logic Before proving that119868(119901 119902) is defined as

119868 (119901 119902) =

1 minus 119865 (119901 119902)norm

119901 gt 119902

1 119901 le 119902

(3)

and 119865(119901 119902)norm is from (2) let us show somebasic operations

in proposed fuzzy logic Let us designate the truth valuesof logical antecedent 119875 and consequent 119876 as 119879(119875) = 119901 and119879(119876) = 119902 respectively Then relevant set of proposed fuzzy

Table 1

Name Designation ValueTautology 119875

119868 1Controversy 119875

119874 0Negation not119875 1 minus 119875

Disjunction 119875 or 119876

119901 + 119902

2

119901 + 119902 lt 1

1 119901 + 119902 ge 1

Conjunction 119875 and 119876

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

Implication 119875 rarr 119876

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 le 119902

Equivalence 119875 harr 119876

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

Pierce arrow 119875 darr 119876

1 minus

119901 + 119902

2

119901 + 119902 lt 1

0 119901 + 119902 ge 1

Shaffer stroke 119875 uarr 119876

1 minus

119901 + 119902

2

119901 + 119902 gt 1

1 119901 + 119902 le 1

logic operators is shown in Table 1 To get the truth values ofthese definitions we use well-known logical properties suchas 119901 rarr 119902 = not119901 or 119902 119901 and 119902 = not(not119901 or not119902) and the like

In other words in [5] we proposed a new many-valuedsystem characterized by the set of base union (cup) andintersection (cap) operations with relevant complement definedas 119879(not119875) = 1 minus 119879(119875) In addition the operators darr and uarr areexpressed as negations of the cup and cap correspondingly Forthis matter let us pose the problem very explicitly

We areworking inmany-valued systemwhich for presentpurposes is all or some of the real intervalR = [0 1] As wasmentioned in [5] the rationales there aremore than ample forall practical purposes the following set 0 01 02 09 1of 11 values is quite sufficient and we will use this set 119881

11in

our illustration Table 2 shows the operation implication inproposed fuzzy logic

32 Geometric Primitives as Fuzzy Predicates It is well knownthat in Boolean predicate logic atomic statements are formal-ized by predicates Predicates that are used in the theory ofincidence geometry may be denoted by 119901(119886) (ldquoa is a pointrdquo)119897(119886) (ldquoa is a linerdquo) and inc(119886 119887) (ldquoa and b are incidentrdquo) Thepredicate expressing equality can be denoted by eq(119886 119887) (ldquoaand b are equalrdquo) Traditionally predicates are interpreted bycrisp relations For example eq119873times119873 rarr 0 1 is a functionthat assigns 1 to every pair of equal objects and 0 to everypair of distinct objects from the set N Of course predicates


ch(A)

c(A)

A

120579max(A)

120579min(A)

(a)

A


120579max(B)

120579max(A)

(b)


Table 2

119901 rarr 119902 0 01 02 03 04 05 06 07 08 09 10 1 1 1 1 1 1 1 1 1 1 101 045 1 1 1 1 1 1 1 1 1 102 04 045 1 1 1 1 1 1 1 1 103 035 04 045 1 1 1 1 1 1 1 104 03 035 04 045 1 1 1 1 1 1 105 025 03 035 04 045 1 1 1 1 1 106 02 025 03 035 04 045 1 1 1 1 107 015 02 025 03 035 04 045 1 1 1 108 01 015 02 025 03 035 04 045 1 1 109 005 01 015 02 025 03 035 04 045 1 11 0 005 01 015 02 025 03 035 04 045 1




120579min (119860)

= min119905

10038161003816100381610038161003816ch (119860) cap 119888 (119860) + 119905 sdot 119877120572 sdot (0 1)

119879| 119905 isin R

10038161003816100381610038161003816

120579max (119860)

= max119905

10038161003816100381610038161003816ch (119860) cap 119888 (119860) + 119905 sdot 119877120572 sdot (0 1)

119879| 119905 isin R

10038161003816100381610038161003816

(4)





119901 (119860) =

120579min (119860)

120579max (119860) (5)


119897 (119860) = 1 minus 119901 (119860) (6)



inc (119860 119861)

= max(|ch (119860) cap ch (119861)||ch (119860)|

|ch (119860) cap ch (119861)||ch (119861)|

)

(7)



eq (119860 119861)

= min(|ch (119860) cap ch (119861)||ch (119860)|

|ch (119860) cap ch (119861)||ch (119861)|

)

(8)





120579max (ch (119860 cup 119861))) (9)



119860 and 119861 =

119886 + 119887

2

119886 + 119887 gt 1

0 119886 + 119887 le 1

(10)

119860 997888rarr 119861 =

1 minus 119886 + 119887

2

119886 gt 119887

1 119886 le 119887

(11)


119860 or 119861 =

119886 + 119887

2

119886 + 119887 lt 1

1 119886 + 119887 ge 1

(12)



inc (119860 119861) =

119886 + 119887

2119886

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(13)


inc (119860 119861) = |119860 cap 119861|

min (|119860| |119861|) (14)

And given that

min (|119860| |119861|) = 119886 + 119887 minus |119886 minus 119887|2

(15)



inc (119860 119861)

=

1 119886 + 119887 gt 1 119886 = 119887 119886 gt 05 119887 gt 05

0 119886 + 119887 le 1

(16)


eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(17)


eq (119860 119861) = min(119860 cap 119861119860

119860 cap 119861

119861

) (18)


119875 =

119886 + 119887

2119886

119886 + 119887 gt 1

0 119886 + 119887 le 1

119876 =

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(19)


find the following

119875 + 119876 =

119886 + 119887

2119886

+

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

=

(119886 + 119887)2

2119886119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(20)


119875 minus 119876 =

119886 + 119887

2119886

minus

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

=

1198872minus 1198862

2119886119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(21)



|119875 minus 119876| =

1198872minus 1198862

2119886119887

119886 + 119887 gt 1 119887 gt 119886

1198862minus 1198872

2119886119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(22)



=

(119886 + 119887)2minus 1198872+ 1198862

2119886119887

119886 + 119887 gt 1 119887 gt 119886

(119886 + 119887)2minus 1198862+ 1198872

4119886119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

=

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(23)


eq (119860 119862) 997888rarr eq (119860 119861) and eq (119861 119862) (24)



eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

eq (119861 119862) =

119887 + 119888

2119888

119887 + 119888 gt 1 119887 lt 119888

119887 + 119888

2119887

119887 + 119888 gt 1 119887 gt 119888

0 119887 + 119888 le 1

(25)

then

eq (119860 119861) and eq (119861 119862)

=

eq (119860 119861) + eq (119861 119862)2

eq (119860 119861) + eq (119861 119862) gt 1

0 eq (119860 119861) + eq (119861 119862) le 1

(26)


eq (119860 119862) =

119886 + 119888

2119888

119886 + 119888 gt 1 119886 lt 119888

119886 + 119888

2119886

119886 + 119888 gt 1 119886 gt 119888

0 119886 + 119888 le 1

(27)


(eq (119860 119861) and eq (119861 119862))2

=

119886 + 119887

2119887

+

119887 + 119888

2119888

=

119886119888 + 2119887119888 + 1198872

4119887119888

(28)


119886 + 119888

2119888

997888rarr

119886119888 + 2119887119888 + 1198872

4119887119888

(29)



2


2= 1


(eq (119860 119861) and eq (119861 119862))2

=

119886 + 119887

2119886

+

119887 + 119888

2119887

=

119886119888 + 2119886119887 + 1198872

4119886119887

(30)


119886 + 119888

2119886

997888rarr

119886119888 + 2119886119887 + 1198872

4119886119887

(31)



2


2= 1


dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(32)




(33)


dp (119860 119861)

=

max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

119886 + 119887 lt 1

max0 1 minus 119886 + 119887 + |119886 minus 119887|2

119886 + 119887 ge 1

(34)


max0 1 minus 119886 + 119887 + |119886 minus 119887|2


=

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

(35)


max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

= max0 minus |119886 minus 119887|119886 + 119887

= 0 119886 + 119887 lt 1

(36)


dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(37)


(11986811015840) dp(119886 119887) rarr sup

119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]


1015840) rarr



inc(119887 119888)(11986841015840) sup119886119887119888119889




1015840) is ful-


Proof From (16)

inc (119860 119862)

=

1 119886 + 119888 gt 1 119886 = 119888 119886 gt 05 119888 gt 05

0 119886 + 119888 le 1

inc (119861 119862)

=

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1


equiv 1

(38)


119888[119897(119888) and 1] =


119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]




inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) (39)


eq (119862 1198621015840) =

119888 + 1198881015840

21198881015840 119888 + 119888

1015840gt 1 119888 lt 119888

1015840

119888 + 1198881015840

2119888

119888 + 1198881015840gt 1 119888 gt 119888

1015840

0 119888 + 1198881015840le 1

(40)

And from (16)

inc (119861 119862) =

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1

(41)


inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) equiv 1 (42)















1015840) is




equiv 1 (48)


eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(49)


noteq (119860 119861) =

119887 minus 119886

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 minus 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(50)


noteq (119860 119861) and 1

=

[1 + (119887 minus 119886) 2119887]

2

119886 + 119887 gt 1 119886 lt 119887

[1 + (119886 minus 119887) 2119886]

2

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

=

3119887 minus 119886

4119887

119886 + 119887 gt 1 119886 lt 119887

3119886 minus 119887

4119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(51)



1 and 05 = 075





L

M

Q

P



inc (119860119863)

=

1 119886 + 119889 gt 1 119886 = 119889 119886 gt 05 119889 gt 05

0 119886 + 119889 le 1

inc (119861 119863)

=

1 119887 + 119889 gt 1 119887 = 119889 119887 gt 05 119889 gt 05

0 119887 + 119889 le 1

inc (119862119863)

=

1 119888 + 119889 gt 1 119888 = 119889 119888 gt 05 119889 gt 05

0 119888 + 119889 le 1

(52)



equiv 1 (53)


119886119887119888119889[119901(119886) and 119901(119887) and 119901(119888) and 1] equiv 1











forall119901 119902 isin [0 1]

dArr

119889 (119901 119901) = 0

119889 (119901 119902) = 119889 (119902 119901)

119889 (119901 119902) + 119889 (119902 119903) ge 119889 (119901 119903)

(55)


119889 (119901 119902) = 0 lArrrArr

119901 = 119902

(56)


372 The t-Norm





119891 (119901 119902) = 119875 cap 119876 =

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

119891 (119902 119901) = 119876 cap 119875 =

119902 + 119901

2

119902 + 119901 gt 1

0 119902 + 119901 le 1

(57)

therefore 119891(119901 119902) = 119891(119902 119901)

Associativity


119891 (119901 119902) =

119901 + 119902

2

119901 + 119902 gt 1 997904rArr

119891 (119901 119902) and 119903 =

119891 (119901 119902) + 119903

2

119891 (119901 119902) + 119903 gt 1

0 119891 (119901 119902) + 119903 le 1

=

119901 + 119902 + 2119903

4

119901 + 119902

2

+ 119903 gt 1

0

119901 + 119902

2

+ 119903 le 1

(58)


1198911(119901 119903) =

119901 + 119902 + 2119903

4

119901 + 119902 + 2119903 gt 2

0 119901 + 119902 + 2119903 le 2

(59)



(59)

1198912(119901 119903) =

119902 + 119903 + 2119901

4

119902 + 119903 + 2119901 gt 2

0 119902 + 119903 + 2119901 le 2

(60)


2(119901 119903) = 0 |


2(119901 119903)


119901 and 119903 =

119901 + 119903

2

119901 + 119903 gt 1

0 119901 + 119903 le 1

119902 and 119903 =

119902 + 119903

2

119902 + 119903 gt 1

0 119902 + 119903 le 1

(61)



Neutrality Consider

1 and 119901 =

1 + 119901

2

1 + 119901 gt 1

0 1 + 119901 le 1

=

1 + 119901

2

119901 gt 0

0 119901 le 0

=

1 + 119901

2

119901 gt 0

0 119901 = 0

(62)


1 and 119901 =

1 + 119901

2

119901 isin (0 1)

119901 119901 = 0 119901 = 1

(63)

Absorption Consider

0 and 119901 =

119901

2

119901 gt 1

0 119901 le 1

(64)



forall119901 119902 isin [0 1]

dArr

119890 (119901 119901) = 1

119890 (119901 119902) = 119890 (119902 119901)

119890 (119901 119902) and 119890 (119902 119903) le 119890 (119901 119903)

(65)


119890 (119901 119902) = 119875 larrrarr

119876 =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(66)


Proof




119890 (119901 119902) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(67)

but

119890 (119902 119901) =

1 minus 119902 + 119901

2

119902 gt 119901

1 119902 = 119901

1 minus 119901 + 119902

2

119902 lt 119901

(68)

therefore 119890(119901 119902) equiv 119890(119902 119901)


From (66) let

1198651(119901 119903) = 119890 (119901 119903) =

1 minus 119901 + 119903

2

119901 gt 119903

1 119901 = 119903

1 minus 119903 + 119901

2

119901 lt 119903

(69)

119890 (119902 119903) =

1 minus 119902 + 119903

2

119902 gt 119903

1 119902 = 119903

1 minus 119903 + 119902

2

119902 lt 119903

(70)

then

1198652(119901 119903) = 119890 (119901 119902) and 119890 (119902 119903)

=

119890 (119901 119902) + 119890 (119902 119903)

2

119890 (119901 119902) + 119890 (119902 119903) gt 1

0 119890 (119901 119902) + 119890 (119902 119903) le 1

(71)

But

1198652(119901 119903) =

119890 (119901 119902) + 119890 (119902 119903)

2

=

((1 minus 119901 + 119902) 2 + (1 minus 119902 + 119903) 2)

2

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

((1 minus 119902 + 119901) 2 + (1 minus 119903 + 119902) 2)

2

119901 lt 119902 lt 119903

=

2 minus 119901 + 119903

4

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

2 minus 119903 + 119901

4

119901 lt 119903 lt 119903

(72)



2(119901 119903) = 119890(119901 119902) and 119890(119902 119903) equiv 0 119890(119901 119902) + 119890(119902 119903) le 1 we


1(119901 119903) hArr



119890 (119901 119902) = 1 minus 119889 (119901 119902) (73)


119889 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

=

1 +1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

119901 = 119902

0 119901 = 119902

(74)



119890 (119901 119902) and 119890 (119902 119903) and dis (119902) le 119890 (119901 119903) (75)



L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)



1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865


spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)


forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr


4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)



holds


120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)


119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)



120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ch(A)

c(A)

A

120579max(A)

120579min(A)

(a)

A


120579max(B)

120579max(A)

(b)


Table 2

119901 rarr 119902 0 01 02 03 04 05 06 07 08 09 10 1 1 1 1 1 1 1 1 1 1 101 045 1 1 1 1 1 1 1 1 1 102 04 045 1 1 1 1 1 1 1 1 103 035 04 045 1 1 1 1 1 1 1 104 03 035 04 045 1 1 1 1 1 1 105 025 03 035 04 045 1 1 1 1 1 106 02 025 03 035 04 045 1 1 1 1 107 015 02 025 03 035 04 045 1 1 1 108 01 015 02 025 03 035 04 045 1 1 109 005 01 015 02 025 03 035 04 045 1 11 0 005 01 015 02 025 03 035 04 045 1




120579min (119860)

= min119905

10038161003816100381610038161003816ch (119860) cap 119888 (119860) + 119905 sdot 119877120572 sdot (0 1)

119879| 119905 isin R

10038161003816100381610038161003816

120579max (119860)

= max119905

10038161003816100381610038161003816ch (119860) cap 119888 (119860) + 119905 sdot 119877120572 sdot (0 1)

119879| 119905 isin R

10038161003816100381610038161003816

(4)





119901 (119860) =

120579min (119860)

120579max (119860) (5)


119897 (119860) = 1 minus 119901 (119860) (6)



inc (119860 119861)

= max(|ch (119860) cap ch (119861)||ch (119860)|

|ch (119860) cap ch (119861)||ch (119861)|

)

(7)



eq (119860 119861)

= min(|ch (119860) cap ch (119861)||ch (119860)|

|ch (119860) cap ch (119861)||ch (119861)|

)

(8)





120579max (ch (119860 cup 119861))) (9)



119860 and 119861 =

119886 + 119887

2

119886 + 119887 gt 1

0 119886 + 119887 le 1

(10)

119860 997888rarr 119861 =

1 minus 119886 + 119887

2

119886 gt 119887

1 119886 le 119887

(11)


119860 or 119861 =

119886 + 119887

2

119886 + 119887 lt 1

1 119886 + 119887 ge 1

(12)



inc (119860 119861) =

119886 + 119887

2119886

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(13)


inc (119860 119861) = |119860 cap 119861|

min (|119860| |119861|) (14)

And given that

min (|119860| |119861|) = 119886 + 119887 minus |119886 minus 119887|2

(15)



inc (119860 119861)

=

1 119886 + 119887 gt 1 119886 = 119887 119886 gt 05 119887 gt 05

0 119886 + 119887 le 1

(16)


eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(17)


eq (119860 119861) = min(119860 cap 119861119860

119860 cap 119861

119861

) (18)


119875 =

119886 + 119887

2119886

119886 + 119887 gt 1

0 119886 + 119887 le 1

119876 =

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(19)


find the following

119875 + 119876 =

119886 + 119887

2119886

+

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

=

(119886 + 119887)2

2119886119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(20)


119875 minus 119876 =

119886 + 119887

2119886

minus

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

=

1198872minus 1198862

2119886119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(21)



|119875 minus 119876| =

1198872minus 1198862

2119886119887

119886 + 119887 gt 1 119887 gt 119886

1198862minus 1198872

2119886119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(22)



=

(119886 + 119887)2minus 1198872+ 1198862

2119886119887

119886 + 119887 gt 1 119887 gt 119886

(119886 + 119887)2minus 1198862+ 1198872

4119886119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

=

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(23)


eq (119860 119862) 997888rarr eq (119860 119861) and eq (119861 119862) (24)



eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

eq (119861 119862) =

119887 + 119888

2119888

119887 + 119888 gt 1 119887 lt 119888

119887 + 119888

2119887

119887 + 119888 gt 1 119887 gt 119888

0 119887 + 119888 le 1

(25)

then

eq (119860 119861) and eq (119861 119862)

=

eq (119860 119861) + eq (119861 119862)2

eq (119860 119861) + eq (119861 119862) gt 1

0 eq (119860 119861) + eq (119861 119862) le 1

(26)


eq (119860 119862) =

119886 + 119888

2119888

119886 + 119888 gt 1 119886 lt 119888

119886 + 119888

2119886

119886 + 119888 gt 1 119886 gt 119888

0 119886 + 119888 le 1

(27)


(eq (119860 119861) and eq (119861 119862))2

=

119886 + 119887

2119887

+

119887 + 119888

2119888

=

119886119888 + 2119887119888 + 1198872

4119887119888

(28)


119886 + 119888

2119888

997888rarr

119886119888 + 2119887119888 + 1198872

4119887119888

(29)



2


2= 1


(eq (119860 119861) and eq (119861 119862))2

=

119886 + 119887

2119886

+

119887 + 119888

2119887

=

119886119888 + 2119886119887 + 1198872

4119886119887

(30)


119886 + 119888

2119886

997888rarr

119886119888 + 2119886119887 + 1198872

4119886119887

(31)



2


2= 1


dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(32)




(33)


dp (119860 119861)

=

max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

119886 + 119887 lt 1

max0 1 minus 119886 + 119887 + |119886 minus 119887|2

119886 + 119887 ge 1

(34)


max0 1 minus 119886 + 119887 + |119886 minus 119887|2


=

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

(35)


max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

= max0 minus |119886 minus 119887|119886 + 119887

= 0 119886 + 119887 lt 1

(36)


dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(37)


(11986811015840) dp(119886 119887) rarr sup

119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]


1015840) rarr



inc(119887 119888)(11986841015840) sup119886119887119888119889




1015840) is ful-


Proof From (16)

inc (119860 119862)

=

1 119886 + 119888 gt 1 119886 = 119888 119886 gt 05 119888 gt 05

0 119886 + 119888 le 1

inc (119861 119862)

=

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1


equiv 1

(38)


119888[119897(119888) and 1] =


119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]




inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) (39)


eq (119862 1198621015840) =

119888 + 1198881015840

21198881015840 119888 + 119888

1015840gt 1 119888 lt 119888

1015840

119888 + 1198881015840

2119888

119888 + 1198881015840gt 1 119888 gt 119888

1015840

0 119888 + 1198881015840le 1

(40)

And from (16)

inc (119861 119862) =

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1

(41)


inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) equiv 1 (42)















1015840) is




equiv 1 (48)


eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(49)


noteq (119860 119861) =

119887 minus 119886

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 minus 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(50)


noteq (119860 119861) and 1

=

[1 + (119887 minus 119886) 2119887]

2

119886 + 119887 gt 1 119886 lt 119887

[1 + (119886 minus 119887) 2119886]

2

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

=

3119887 minus 119886

4119887

119886 + 119887 gt 1 119886 lt 119887

3119886 minus 119887

4119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(51)



1 and 05 = 075





L

M

Q

P



inc (119860119863)

=

1 119886 + 119889 gt 1 119886 = 119889 119886 gt 05 119889 gt 05

0 119886 + 119889 le 1

inc (119861 119863)

=

1 119887 + 119889 gt 1 119887 = 119889 119887 gt 05 119889 gt 05

0 119887 + 119889 le 1

inc (119862119863)

=

1 119888 + 119889 gt 1 119888 = 119889 119888 gt 05 119889 gt 05

0 119888 + 119889 le 1

(52)



equiv 1 (53)


119886119887119888119889[119901(119886) and 119901(119887) and 119901(119888) and 1] equiv 1











forall119901 119902 isin [0 1]

dArr

119889 (119901 119901) = 0

119889 (119901 119902) = 119889 (119902 119901)

119889 (119901 119902) + 119889 (119902 119903) ge 119889 (119901 119903)

(55)


119889 (119901 119902) = 0 lArrrArr

119901 = 119902

(56)


372 The t-Norm





119891 (119901 119902) = 119875 cap 119876 =

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

119891 (119902 119901) = 119876 cap 119875 =

119902 + 119901

2

119902 + 119901 gt 1

0 119902 + 119901 le 1

(57)

therefore 119891(119901 119902) = 119891(119902 119901)

Associativity


119891 (119901 119902) =

119901 + 119902

2

119901 + 119902 gt 1 997904rArr

119891 (119901 119902) and 119903 =

119891 (119901 119902) + 119903

2

119891 (119901 119902) + 119903 gt 1

0 119891 (119901 119902) + 119903 le 1

=

119901 + 119902 + 2119903

4

119901 + 119902

2

+ 119903 gt 1

0

119901 + 119902

2

+ 119903 le 1

(58)


1198911(119901 119903) =

119901 + 119902 + 2119903

4

119901 + 119902 + 2119903 gt 2

0 119901 + 119902 + 2119903 le 2

(59)



(59)

1198912(119901 119903) =

119902 + 119903 + 2119901

4

119902 + 119903 + 2119901 gt 2

0 119902 + 119903 + 2119901 le 2

(60)


2(119901 119903) = 0 |


2(119901 119903)


119901 and 119903 =

119901 + 119903

2

119901 + 119903 gt 1

0 119901 + 119903 le 1

119902 and 119903 =

119902 + 119903

2

119902 + 119903 gt 1

0 119902 + 119903 le 1

(61)



Neutrality Consider

1 and 119901 =

1 + 119901

2

1 + 119901 gt 1

0 1 + 119901 le 1

=

1 + 119901

2

119901 gt 0

0 119901 le 0

=

1 + 119901

2

119901 gt 0

0 119901 = 0

(62)


1 and 119901 =

1 + 119901

2

119901 isin (0 1)

119901 119901 = 0 119901 = 1

(63)

Absorption Consider

0 and 119901 =

119901

2

119901 gt 1

0 119901 le 1

(64)



forall119901 119902 isin [0 1]

dArr

119890 (119901 119901) = 1

119890 (119901 119902) = 119890 (119902 119901)

119890 (119901 119902) and 119890 (119902 119903) le 119890 (119901 119903)

(65)


119890 (119901 119902) = 119875 larrrarr

119876 =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(66)


Proof




119890 (119901 119902) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(67)

but

119890 (119902 119901) =

1 minus 119902 + 119901

2

119902 gt 119901

1 119902 = 119901

1 minus 119901 + 119902

2

119902 lt 119901

(68)

therefore 119890(119901 119902) equiv 119890(119902 119901)


From (66) let

1198651(119901 119903) = 119890 (119901 119903) =

1 minus 119901 + 119903

2

119901 gt 119903

1 119901 = 119903

1 minus 119903 + 119901

2

119901 lt 119903

(69)

119890 (119902 119903) =

1 minus 119902 + 119903

2

119902 gt 119903

1 119902 = 119903

1 minus 119903 + 119902

2

119902 lt 119903

(70)

then

1198652(119901 119903) = 119890 (119901 119902) and 119890 (119902 119903)

=

119890 (119901 119902) + 119890 (119902 119903)

2

119890 (119901 119902) + 119890 (119902 119903) gt 1

0 119890 (119901 119902) + 119890 (119902 119903) le 1

(71)

But

1198652(119901 119903) =

119890 (119901 119902) + 119890 (119902 119903)

2

=

((1 minus 119901 + 119902) 2 + (1 minus 119902 + 119903) 2)

2

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

((1 minus 119902 + 119901) 2 + (1 minus 119903 + 119902) 2)

2

119901 lt 119902 lt 119903

=

2 minus 119901 + 119903

4

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

2 minus 119903 + 119901

4

119901 lt 119903 lt 119903

(72)



2(119901 119903) = 119890(119901 119902) and 119890(119902 119903) equiv 0 119890(119901 119902) + 119890(119902 119903) le 1 we


1(119901 119903) hArr



119890 (119901 119902) = 1 minus 119889 (119901 119902) (73)


119889 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

=

1 +1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

119901 = 119902

0 119901 = 119902

(74)



119890 (119901 119902) and 119890 (119902 119903) and dis (119902) le 119890 (119901 119903) (75)



L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)



1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865


spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)


forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr


4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)



holds


120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)


119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)



120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120579max (ch (119860 cup 119861))) (9)



119860 and 119861 =

119886 + 119887

2

119886 + 119887 gt 1

0 119886 + 119887 le 1

(10)

119860 997888rarr 119861 =

1 minus 119886 + 119887

2

119886 gt 119887

1 119886 le 119887

(11)


119860 or 119861 =

119886 + 119887

2

119886 + 119887 lt 1

1 119886 + 119887 ge 1

(12)



inc (119860 119861) =

119886 + 119887

2119886

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(13)


inc (119860 119861) = |119860 cap 119861|

min (|119860| |119861|) (14)

And given that

min (|119860| |119861|) = 119886 + 119887 minus |119886 minus 119887|2

(15)



inc (119860 119861)

=

1 119886 + 119887 gt 1 119886 = 119887 119886 gt 05 119887 gt 05

0 119886 + 119887 le 1

(16)


eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(17)


eq (119860 119861) = min(119860 cap 119861119860

119860 cap 119861

119861

) (18)


119875 =

119886 + 119887

2119886

119886 + 119887 gt 1

0 119886 + 119887 le 1

119876 =

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(19)


find the following

119875 + 119876 =

119886 + 119887

2119886

+

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

=

(119886 + 119887)2

2119886119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(20)


119875 minus 119876 =

119886 + 119887

2119886

minus

119886 + 119887

2119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

=

1198872minus 1198862

2119886119887

119886 + 119887 gt 1

0 119886 + 119887 le 1

(21)



|119875 minus 119876| =

1198872minus 1198862

2119886119887

119886 + 119887 gt 1 119887 gt 119886

1198862minus 1198872

2119886119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(22)



=

(119886 + 119887)2minus 1198872+ 1198862

2119886119887

119886 + 119887 gt 1 119887 gt 119886

(119886 + 119887)2minus 1198862+ 1198872

4119886119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

=

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(23)


eq (119860 119862) 997888rarr eq (119860 119861) and eq (119861 119862) (24)



eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

eq (119861 119862) =

119887 + 119888

2119888

119887 + 119888 gt 1 119887 lt 119888

119887 + 119888

2119887

119887 + 119888 gt 1 119887 gt 119888

0 119887 + 119888 le 1

(25)

then

eq (119860 119861) and eq (119861 119862)

=

eq (119860 119861) + eq (119861 119862)2

eq (119860 119861) + eq (119861 119862) gt 1

0 eq (119860 119861) + eq (119861 119862) le 1

(26)


eq (119860 119862) =

119886 + 119888

2119888

119886 + 119888 gt 1 119886 lt 119888

119886 + 119888

2119886

119886 + 119888 gt 1 119886 gt 119888

0 119886 + 119888 le 1

(27)


(eq (119860 119861) and eq (119861 119862))2

=

119886 + 119887

2119887

+

119887 + 119888

2119888

=

119886119888 + 2119887119888 + 1198872

4119887119888

(28)


119886 + 119888

2119888

997888rarr

119886119888 + 2119887119888 + 1198872

4119887119888

(29)



2


2= 1


(eq (119860 119861) and eq (119861 119862))2

=

119886 + 119887

2119886

+

119887 + 119888

2119887

=

119886119888 + 2119886119887 + 1198872

4119886119887

(30)


119886 + 119888

2119886

997888rarr

119886119888 + 2119886119887 + 1198872

4119886119887

(31)



2


2= 1


dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(32)




(33)


dp (119860 119861)

=

max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

119886 + 119887 lt 1

max0 1 minus 119886 + 119887 + |119886 minus 119887|2

119886 + 119887 ge 1

(34)


max0 1 minus 119886 + 119887 + |119886 minus 119887|2


=

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

(35)


max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

= max0 minus |119886 minus 119887|119886 + 119887

= 0 119886 + 119887 lt 1

(36)


dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(37)


(11986811015840) dp(119886 119887) rarr sup

119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]


1015840) rarr



inc(119887 119888)(11986841015840) sup119886119887119888119889




1015840) is ful-


Proof From (16)

inc (119860 119862)

=

1 119886 + 119888 gt 1 119886 = 119888 119886 gt 05 119888 gt 05

0 119886 + 119888 le 1

inc (119861 119862)

=

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1


equiv 1

(38)


119888[119897(119888) and 1] =


119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]




inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) (39)


eq (119862 1198621015840) =

119888 + 1198881015840

21198881015840 119888 + 119888

1015840gt 1 119888 lt 119888

1015840

119888 + 1198881015840

2119888

119888 + 1198881015840gt 1 119888 gt 119888

1015840

0 119888 + 1198881015840le 1

(40)

And from (16)

inc (119861 119862) =

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1

(41)


inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) equiv 1 (42)















1015840) is




equiv 1 (48)


eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(49)


noteq (119860 119861) =

119887 minus 119886

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 minus 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(50)


noteq (119860 119861) and 1

=

[1 + (119887 minus 119886) 2119887]

2

119886 + 119887 gt 1 119886 lt 119887

[1 + (119886 minus 119887) 2119886]

2

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

=

3119887 minus 119886

4119887

119886 + 119887 gt 1 119886 lt 119887

3119886 minus 119887

4119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(51)



1 and 05 = 075





L

M

Q

P



inc (119860119863)

=

1 119886 + 119889 gt 1 119886 = 119889 119886 gt 05 119889 gt 05

0 119886 + 119889 le 1

inc (119861 119863)

=

1 119887 + 119889 gt 1 119887 = 119889 119887 gt 05 119889 gt 05

0 119887 + 119889 le 1

inc (119862119863)

=

1 119888 + 119889 gt 1 119888 = 119889 119888 gt 05 119889 gt 05

0 119888 + 119889 le 1

(52)



equiv 1 (53)


119886119887119888119889[119901(119886) and 119901(119887) and 119901(119888) and 1] equiv 1











forall119901 119902 isin [0 1]

dArr

119889 (119901 119901) = 0

119889 (119901 119902) = 119889 (119902 119901)

119889 (119901 119902) + 119889 (119902 119903) ge 119889 (119901 119903)

(55)


119889 (119901 119902) = 0 lArrrArr

119901 = 119902

(56)


372 The t-Norm





119891 (119901 119902) = 119875 cap 119876 =

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

119891 (119902 119901) = 119876 cap 119875 =

119902 + 119901

2

119902 + 119901 gt 1

0 119902 + 119901 le 1

(57)

therefore 119891(119901 119902) = 119891(119902 119901)

Associativity


119891 (119901 119902) =

119901 + 119902

2

119901 + 119902 gt 1 997904rArr

119891 (119901 119902) and 119903 =

119891 (119901 119902) + 119903

2

119891 (119901 119902) + 119903 gt 1

0 119891 (119901 119902) + 119903 le 1

=

119901 + 119902 + 2119903

4

119901 + 119902

2

+ 119903 gt 1

0

119901 + 119902

2

+ 119903 le 1

(58)


1198911(119901 119903) =

119901 + 119902 + 2119903

4

119901 + 119902 + 2119903 gt 2

0 119901 + 119902 + 2119903 le 2

(59)



(59)

1198912(119901 119903) =

119902 + 119903 + 2119901

4

119902 + 119903 + 2119901 gt 2

0 119902 + 119903 + 2119901 le 2

(60)


2(119901 119903) = 0 |


2(119901 119903)


119901 and 119903 =

119901 + 119903

2

119901 + 119903 gt 1

0 119901 + 119903 le 1

119902 and 119903 =

119902 + 119903

2

119902 + 119903 gt 1

0 119902 + 119903 le 1

(61)



Neutrality Consider

1 and 119901 =

1 + 119901

2

1 + 119901 gt 1

0 1 + 119901 le 1

=

1 + 119901

2

119901 gt 0

0 119901 le 0

=

1 + 119901

2

119901 gt 0

0 119901 = 0

(62)


1 and 119901 =

1 + 119901

2

119901 isin (0 1)

119901 119901 = 0 119901 = 1

(63)

Absorption Consider

0 and 119901 =

119901

2

119901 gt 1

0 119901 le 1

(64)



forall119901 119902 isin [0 1]

dArr

119890 (119901 119901) = 1

119890 (119901 119902) = 119890 (119902 119901)

119890 (119901 119902) and 119890 (119902 119903) le 119890 (119901 119903)

(65)


119890 (119901 119902) = 119875 larrrarr

119876 =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(66)


Proof




119890 (119901 119902) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(67)

but

119890 (119902 119901) =

1 minus 119902 + 119901

2

119902 gt 119901

1 119902 = 119901

1 minus 119901 + 119902

2

119902 lt 119901

(68)

therefore 119890(119901 119902) equiv 119890(119902 119901)


From (66) let

1198651(119901 119903) = 119890 (119901 119903) =

1 minus 119901 + 119903

2

119901 gt 119903

1 119901 = 119903

1 minus 119903 + 119901

2

119901 lt 119903

(69)

119890 (119902 119903) =

1 minus 119902 + 119903

2

119902 gt 119903

1 119902 = 119903

1 minus 119903 + 119902

2

119902 lt 119903

(70)

then

1198652(119901 119903) = 119890 (119901 119902) and 119890 (119902 119903)

=

119890 (119901 119902) + 119890 (119902 119903)

2

119890 (119901 119902) + 119890 (119902 119903) gt 1

0 119890 (119901 119902) + 119890 (119902 119903) le 1

(71)

But

1198652(119901 119903) =

119890 (119901 119902) + 119890 (119902 119903)

2

=

((1 minus 119901 + 119902) 2 + (1 minus 119902 + 119903) 2)

2

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

((1 minus 119902 + 119901) 2 + (1 minus 119903 + 119902) 2)

2

119901 lt 119902 lt 119903

=

2 minus 119901 + 119903

4

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

2 minus 119903 + 119901

4

119901 lt 119903 lt 119903

(72)



2(119901 119903) = 119890(119901 119902) and 119890(119902 119903) equiv 0 119890(119901 119902) + 119890(119902 119903) le 1 we


1(119901 119903) hArr



119890 (119901 119902) = 1 minus 119889 (119901 119902) (73)


119889 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

=

1 +1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

119901 = 119902

0 119901 = 119902

(74)



119890 (119901 119902) and 119890 (119902 119903) and dis (119902) le 119890 (119901 119903) (75)



L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)



1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865


spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)


forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr


4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)



holds


120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)


119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)



120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











|119875 minus 119876| =

1198872minus 1198862

2119886119887

119886 + 119887 gt 1 119887 gt 119886

1198862minus 1198872

2119886119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(22)



=

(119886 + 119887)2minus 1198872+ 1198862

2119886119887

119886 + 119887 gt 1 119887 gt 119886

(119886 + 119887)2minus 1198862+ 1198872

4119886119887

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

=

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(23)


eq (119860 119862) 997888rarr eq (119860 119861) and eq (119861 119862) (24)



eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

eq (119861 119862) =

119887 + 119888

2119888

119887 + 119888 gt 1 119887 lt 119888

119887 + 119888

2119887

119887 + 119888 gt 1 119887 gt 119888

0 119887 + 119888 le 1

(25)

then

eq (119860 119861) and eq (119861 119862)

=

eq (119860 119861) + eq (119861 119862)2

eq (119860 119861) + eq (119861 119862) gt 1

0 eq (119860 119861) + eq (119861 119862) le 1

(26)


eq (119860 119862) =

119886 + 119888

2119888

119886 + 119888 gt 1 119886 lt 119888

119886 + 119888

2119886

119886 + 119888 gt 1 119886 gt 119888

0 119886 + 119888 le 1

(27)


(eq (119860 119861) and eq (119861 119862))2

=

119886 + 119887

2119887

+

119887 + 119888

2119888

=

119886119888 + 2119887119888 + 1198872

4119887119888

(28)


119886 + 119888

2119888

997888rarr

119886119888 + 2119887119888 + 1198872

4119887119888

(29)



2


2= 1


(eq (119860 119861) and eq (119861 119862))2

=

119886 + 119887

2119886

+

119887 + 119888

2119887

=

119886119888 + 2119886119887 + 1198872

4119886119887

(30)


119886 + 119888

2119886

997888rarr

119886119888 + 2119886119887 + 1198872

4119886119887

(31)



2


2= 1


dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(32)




(33)


dp (119860 119861)

=

max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

119886 + 119887 lt 1

max0 1 minus 119886 + 119887 + |119886 minus 119887|2

119886 + 119887 ge 1

(34)


max0 1 minus 119886 + 119887 + |119886 minus 119887|2


=

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

(35)


max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

= max0 minus |119886 minus 119887|119886 + 119887

= 0 119886 + 119887 lt 1

(36)


dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(37)


(11986811015840) dp(119886 119887) rarr sup

119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]


1015840) rarr



inc(119887 119888)(11986841015840) sup119886119887119888119889




1015840) is ful-


Proof From (16)

inc (119860 119862)

=

1 119886 + 119888 gt 1 119886 = 119888 119886 gt 05 119888 gt 05

0 119886 + 119888 le 1

inc (119861 119862)

=

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1


equiv 1

(38)


119888[119897(119888) and 1] =


119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]




inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) (39)


eq (119862 1198621015840) =

119888 + 1198881015840

21198881015840 119888 + 119888

1015840gt 1 119888 lt 119888

1015840

119888 + 1198881015840

2119888

119888 + 1198881015840gt 1 119888 gt 119888

1015840

0 119888 + 1198881015840le 1

(40)

And from (16)

inc (119861 119862) =

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1

(41)


inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) equiv 1 (42)















1015840) is




equiv 1 (48)


eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(49)


noteq (119860 119861) =

119887 minus 119886

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 minus 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(50)


noteq (119860 119861) and 1

=

[1 + (119887 minus 119886) 2119887]

2

119886 + 119887 gt 1 119886 lt 119887

[1 + (119886 minus 119887) 2119886]

2

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

=

3119887 minus 119886

4119887

119886 + 119887 gt 1 119886 lt 119887

3119886 minus 119887

4119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(51)



1 and 05 = 075





L

M

Q

P



inc (119860119863)

=

1 119886 + 119889 gt 1 119886 = 119889 119886 gt 05 119889 gt 05

0 119886 + 119889 le 1

inc (119861 119863)

=

1 119887 + 119889 gt 1 119887 = 119889 119887 gt 05 119889 gt 05

0 119887 + 119889 le 1

inc (119862119863)

=

1 119888 + 119889 gt 1 119888 = 119889 119888 gt 05 119889 gt 05

0 119888 + 119889 le 1

(52)



equiv 1 (53)


119886119887119888119889[119901(119886) and 119901(119887) and 119901(119888) and 1] equiv 1











forall119901 119902 isin [0 1]

dArr

119889 (119901 119901) = 0

119889 (119901 119902) = 119889 (119902 119901)

119889 (119901 119902) + 119889 (119902 119903) ge 119889 (119901 119903)

(55)


119889 (119901 119902) = 0 lArrrArr

119901 = 119902

(56)


372 The t-Norm





119891 (119901 119902) = 119875 cap 119876 =

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

119891 (119902 119901) = 119876 cap 119875 =

119902 + 119901

2

119902 + 119901 gt 1

0 119902 + 119901 le 1

(57)

therefore 119891(119901 119902) = 119891(119902 119901)

Associativity


119891 (119901 119902) =

119901 + 119902

2

119901 + 119902 gt 1 997904rArr

119891 (119901 119902) and 119903 =

119891 (119901 119902) + 119903

2

119891 (119901 119902) + 119903 gt 1

0 119891 (119901 119902) + 119903 le 1

=

119901 + 119902 + 2119903

4

119901 + 119902

2

+ 119903 gt 1

0

119901 + 119902

2

+ 119903 le 1

(58)


1198911(119901 119903) =

119901 + 119902 + 2119903

4

119901 + 119902 + 2119903 gt 2

0 119901 + 119902 + 2119903 le 2

(59)



(59)

1198912(119901 119903) =

119902 + 119903 + 2119901

4

119902 + 119903 + 2119901 gt 2

0 119902 + 119903 + 2119901 le 2

(60)


2(119901 119903) = 0 |


2(119901 119903)


119901 and 119903 =

119901 + 119903

2

119901 + 119903 gt 1

0 119901 + 119903 le 1

119902 and 119903 =

119902 + 119903

2

119902 + 119903 gt 1

0 119902 + 119903 le 1

(61)



Neutrality Consider

1 and 119901 =

1 + 119901

2

1 + 119901 gt 1

0 1 + 119901 le 1

=

1 + 119901

2

119901 gt 0

0 119901 le 0

=

1 + 119901

2

119901 gt 0

0 119901 = 0

(62)


1 and 119901 =

1 + 119901

2

119901 isin (0 1)

119901 119901 = 0 119901 = 1

(63)

Absorption Consider

0 and 119901 =

119901

2

119901 gt 1

0 119901 le 1

(64)



forall119901 119902 isin [0 1]

dArr

119890 (119901 119901) = 1

119890 (119901 119902) = 119890 (119902 119901)

119890 (119901 119902) and 119890 (119902 119903) le 119890 (119901 119903)

(65)


119890 (119901 119902) = 119875 larrrarr

119876 =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(66)


Proof




119890 (119901 119902) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(67)

but

119890 (119902 119901) =

1 minus 119902 + 119901

2

119902 gt 119901

1 119902 = 119901

1 minus 119901 + 119902

2

119902 lt 119901

(68)

therefore 119890(119901 119902) equiv 119890(119902 119901)


From (66) let

1198651(119901 119903) = 119890 (119901 119903) =

1 minus 119901 + 119903

2

119901 gt 119903

1 119901 = 119903

1 minus 119903 + 119901

2

119901 lt 119903

(69)

119890 (119902 119903) =

1 minus 119902 + 119903

2

119902 gt 119903

1 119902 = 119903

1 minus 119903 + 119902

2

119902 lt 119903

(70)

then

1198652(119901 119903) = 119890 (119901 119902) and 119890 (119902 119903)

=

119890 (119901 119902) + 119890 (119902 119903)

2

119890 (119901 119902) + 119890 (119902 119903) gt 1

0 119890 (119901 119902) + 119890 (119902 119903) le 1

(71)

But

1198652(119901 119903) =

119890 (119901 119902) + 119890 (119902 119903)

2

=

((1 minus 119901 + 119902) 2 + (1 minus 119902 + 119903) 2)

2

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

((1 minus 119902 + 119901) 2 + (1 minus 119903 + 119902) 2)

2

119901 lt 119902 lt 119903

=

2 minus 119901 + 119903

4

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

2 minus 119903 + 119901

4

119901 lt 119903 lt 119903

(72)



2(119901 119903) = 119890(119901 119902) and 119890(119902 119903) equiv 0 119890(119901 119902) + 119890(119902 119903) le 1 we


1(119901 119903) hArr



119890 (119901 119902) = 1 minus 119889 (119901 119902) (73)


119889 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

=

1 +1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

119901 = 119902

0 119901 = 119902

(74)



119890 (119901 119902) and 119890 (119902 119903) and dis (119902) le 119890 (119901 119903) (75)



L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)



1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865


spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)


forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr


4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)



holds


120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)


119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)



120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(33)


dp (119860 119861)

=

max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

119886 + 119887 lt 1

max0 1 minus 119886 + 119887 + |119886 minus 119887|2

119886 + 119887 ge 1

(34)


max0 1 minus 119886 + 119887 + |119886 minus 119887|2


=

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

(35)


max0 1 minus 119886 + 119887 + |119886 minus 119887|119886 + 119887

= max0 minus |119886 minus 119887|119886 + 119887

= 0 119886 + 119887 lt 1

(36)


dp (119860 119861) =

1 minus 119886 119886 + 119887 ge 1 119886 ge 119887

1 minus 119887 119886 + 119887 ge 1 119886 lt 119887

0 119886 + 119887 lt 1

(37)


(11986811015840) dp(119886 119887) rarr sup

119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]


1015840) rarr



inc(119887 119888)(11986841015840) sup119886119887119888119889




1015840) is ful-


Proof From (16)

inc (119860 119862)

=

1 119886 + 119888 gt 1 119886 = 119888 119886 gt 05 119888 gt 05

0 119886 + 119888 le 1

inc (119861 119862)

=

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1


equiv 1

(38)


119888[119897(119888) and 1] =


119888[119897(119888) and inc(119886 119888) and inc(119887 119888)]




inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) (39)


eq (119862 1198621015840) =

119888 + 1198881015840

21198881015840 119888 + 119888

1015840gt 1 119888 lt 119888

1015840

119888 + 1198881015840

2119888

119888 + 1198881015840gt 1 119888 gt 119888

1015840

0 119888 + 1198881015840le 1

(40)

And from (16)

inc (119861 119862) =

1 119887 + 119888 gt 1 119887 = 119888 119887 gt 05 119888 gt 05

0 119887 + 119888 le 1

(41)


inc (119887 1198881015840) 997888rarr eq (119888 1198881015840) equiv 1 (42)















1015840) is




equiv 1 (48)


eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(49)


noteq (119860 119861) =

119887 minus 119886

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 minus 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(50)


noteq (119860 119861) and 1

=

[1 + (119887 minus 119886) 2119887]

2

119886 + 119887 gt 1 119886 lt 119887

[1 + (119886 minus 119887) 2119886]

2

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

=

3119887 minus 119886

4119887

119886 + 119887 gt 1 119886 lt 119887

3119886 minus 119887

4119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(51)



1 and 05 = 075





L

M

Q

P



inc (119860119863)

=

1 119886 + 119889 gt 1 119886 = 119889 119886 gt 05 119889 gt 05

0 119886 + 119889 le 1

inc (119861 119863)

=

1 119887 + 119889 gt 1 119887 = 119889 119887 gt 05 119889 gt 05

0 119887 + 119889 le 1

inc (119862119863)

=

1 119888 + 119889 gt 1 119888 = 119889 119888 gt 05 119889 gt 05

0 119888 + 119889 le 1

(52)



equiv 1 (53)


119886119887119888119889[119901(119886) and 119901(119887) and 119901(119888) and 1] equiv 1











forall119901 119902 isin [0 1]

dArr

119889 (119901 119901) = 0

119889 (119901 119902) = 119889 (119902 119901)

119889 (119901 119902) + 119889 (119902 119903) ge 119889 (119901 119903)

(55)


119889 (119901 119902) = 0 lArrrArr

119901 = 119902

(56)


372 The t-Norm





119891 (119901 119902) = 119875 cap 119876 =

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

119891 (119902 119901) = 119876 cap 119875 =

119902 + 119901

2

119902 + 119901 gt 1

0 119902 + 119901 le 1

(57)

therefore 119891(119901 119902) = 119891(119902 119901)

Associativity


119891 (119901 119902) =

119901 + 119902

2

119901 + 119902 gt 1 997904rArr

119891 (119901 119902) and 119903 =

119891 (119901 119902) + 119903

2

119891 (119901 119902) + 119903 gt 1

0 119891 (119901 119902) + 119903 le 1

=

119901 + 119902 + 2119903

4

119901 + 119902

2

+ 119903 gt 1

0

119901 + 119902

2

+ 119903 le 1

(58)


1198911(119901 119903) =

119901 + 119902 + 2119903

4

119901 + 119902 + 2119903 gt 2

0 119901 + 119902 + 2119903 le 2

(59)



(59)

1198912(119901 119903) =

119902 + 119903 + 2119901

4

119902 + 119903 + 2119901 gt 2

0 119902 + 119903 + 2119901 le 2

(60)


2(119901 119903) = 0 |


2(119901 119903)


119901 and 119903 =

119901 + 119903

2

119901 + 119903 gt 1

0 119901 + 119903 le 1

119902 and 119903 =

119902 + 119903

2

119902 + 119903 gt 1

0 119902 + 119903 le 1

(61)



Neutrality Consider

1 and 119901 =

1 + 119901

2

1 + 119901 gt 1

0 1 + 119901 le 1

=

1 + 119901

2

119901 gt 0

0 119901 le 0

=

1 + 119901

2

119901 gt 0

0 119901 = 0

(62)


1 and 119901 =

1 + 119901

2

119901 isin (0 1)

119901 119901 = 0 119901 = 1

(63)

Absorption Consider

0 and 119901 =

119901

2

119901 gt 1

0 119901 le 1

(64)



forall119901 119902 isin [0 1]

dArr

119890 (119901 119901) = 1

119890 (119901 119902) = 119890 (119902 119901)

119890 (119901 119902) and 119890 (119902 119903) le 119890 (119901 119903)

(65)


119890 (119901 119902) = 119875 larrrarr

119876 =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(66)


Proof




119890 (119901 119902) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(67)

but

119890 (119902 119901) =

1 minus 119902 + 119901

2

119902 gt 119901

1 119902 = 119901

1 minus 119901 + 119902

2

119902 lt 119901

(68)

therefore 119890(119901 119902) equiv 119890(119902 119901)


From (66) let

1198651(119901 119903) = 119890 (119901 119903) =

1 minus 119901 + 119903

2

119901 gt 119903

1 119901 = 119903

1 minus 119903 + 119901

2

119901 lt 119903

(69)

119890 (119902 119903) =

1 minus 119902 + 119903

2

119902 gt 119903

1 119902 = 119903

1 minus 119903 + 119902

2

119902 lt 119903

(70)

then

1198652(119901 119903) = 119890 (119901 119902) and 119890 (119902 119903)

=

119890 (119901 119902) + 119890 (119902 119903)

2

119890 (119901 119902) + 119890 (119902 119903) gt 1

0 119890 (119901 119902) + 119890 (119902 119903) le 1

(71)

But

1198652(119901 119903) =

119890 (119901 119902) + 119890 (119902 119903)

2

=

((1 minus 119901 + 119902) 2 + (1 minus 119902 + 119903) 2)

2

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

((1 minus 119902 + 119901) 2 + (1 minus 119903 + 119902) 2)

2

119901 lt 119902 lt 119903

=

2 minus 119901 + 119903

4

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

2 minus 119903 + 119901

4

119901 lt 119903 lt 119903

(72)



2(119901 119903) = 119890(119901 119902) and 119890(119902 119903) equiv 0 119890(119901 119902) + 119890(119902 119903) le 1 we


1(119901 119903) hArr



119890 (119901 119902) = 1 minus 119889 (119901 119902) (73)


119889 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

=

1 +1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

119901 = 119902

0 119901 = 119902

(74)



119890 (119901 119902) and 119890 (119902 119903) and dis (119902) le 119890 (119901 119903) (75)



L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)



1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865


spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)


forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr


4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)



holds


120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)


119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)



120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















1015840) is




equiv 1 (48)


eq (119860 119861) =

119886 + 119887

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 + 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

0 119886 + 119887 le 1

(49)


noteq (119860 119861) =

119887 minus 119886

2119887

119886 + 119887 gt 1 119886 lt 119887

119886 minus 119887

2119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(50)


noteq (119860 119861) and 1

=

[1 + (119887 minus 119886) 2119887]

2

119886 + 119887 gt 1 119886 lt 119887

[1 + (119886 minus 119887) 2119886]

2

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

=

3119887 minus 119886

4119887

119886 + 119887 gt 1 119886 lt 119887

3119886 minus 119887

4119886

119886 + 119887 gt 1 119886 gt 119887

1 119886 + 119887 le 1

(51)



1 and 05 = 075





L

M

Q

P



inc (119860119863)

=

1 119886 + 119889 gt 1 119886 = 119889 119886 gt 05 119889 gt 05

0 119886 + 119889 le 1

inc (119861 119863)

=

1 119887 + 119889 gt 1 119887 = 119889 119887 gt 05 119889 gt 05

0 119887 + 119889 le 1

inc (119862119863)

=

1 119888 + 119889 gt 1 119888 = 119889 119888 gt 05 119889 gt 05

0 119888 + 119889 le 1

(52)



equiv 1 (53)


119886119887119888119889[119901(119886) and 119901(119887) and 119901(119888) and 1] equiv 1











forall119901 119902 isin [0 1]

dArr

119889 (119901 119901) = 0

119889 (119901 119902) = 119889 (119902 119901)

119889 (119901 119902) + 119889 (119902 119903) ge 119889 (119901 119903)

(55)


119889 (119901 119902) = 0 lArrrArr

119901 = 119902

(56)


372 The t-Norm





119891 (119901 119902) = 119875 cap 119876 =

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

119891 (119902 119901) = 119876 cap 119875 =

119902 + 119901

2

119902 + 119901 gt 1

0 119902 + 119901 le 1

(57)

therefore 119891(119901 119902) = 119891(119902 119901)

Associativity


119891 (119901 119902) =

119901 + 119902

2

119901 + 119902 gt 1 997904rArr

119891 (119901 119902) and 119903 =

119891 (119901 119902) + 119903

2

119891 (119901 119902) + 119903 gt 1

0 119891 (119901 119902) + 119903 le 1

=

119901 + 119902 + 2119903

4

119901 + 119902

2

+ 119903 gt 1

0

119901 + 119902

2

+ 119903 le 1

(58)


1198911(119901 119903) =

119901 + 119902 + 2119903

4

119901 + 119902 + 2119903 gt 2

0 119901 + 119902 + 2119903 le 2

(59)



(59)

1198912(119901 119903) =

119902 + 119903 + 2119901

4

119902 + 119903 + 2119901 gt 2

0 119902 + 119903 + 2119901 le 2

(60)


2(119901 119903) = 0 |


2(119901 119903)


119901 and 119903 =

119901 + 119903

2

119901 + 119903 gt 1

0 119901 + 119903 le 1

119902 and 119903 =

119902 + 119903

2

119902 + 119903 gt 1

0 119902 + 119903 le 1

(61)



Neutrality Consider

1 and 119901 =

1 + 119901

2

1 + 119901 gt 1

0 1 + 119901 le 1

=

1 + 119901

2

119901 gt 0

0 119901 le 0

=

1 + 119901

2

119901 gt 0

0 119901 = 0

(62)


1 and 119901 =

1 + 119901

2

119901 isin (0 1)

119901 119901 = 0 119901 = 1

(63)

Absorption Consider

0 and 119901 =

119901

2

119901 gt 1

0 119901 le 1

(64)



forall119901 119902 isin [0 1]

dArr

119890 (119901 119901) = 1

119890 (119901 119902) = 119890 (119902 119901)

119890 (119901 119902) and 119890 (119902 119903) le 119890 (119901 119903)

(65)


119890 (119901 119902) = 119875 larrrarr

119876 =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(66)


Proof




119890 (119901 119902) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(67)

but

119890 (119902 119901) =

1 minus 119902 + 119901

2

119902 gt 119901

1 119902 = 119901

1 minus 119901 + 119902

2

119902 lt 119901

(68)

therefore 119890(119901 119902) equiv 119890(119902 119901)


From (66) let

1198651(119901 119903) = 119890 (119901 119903) =

1 minus 119901 + 119903

2

119901 gt 119903

1 119901 = 119903

1 minus 119903 + 119901

2

119901 lt 119903

(69)

119890 (119902 119903) =

1 minus 119902 + 119903

2

119902 gt 119903

1 119902 = 119903

1 minus 119903 + 119902

2

119902 lt 119903

(70)

then

1198652(119901 119903) = 119890 (119901 119902) and 119890 (119902 119903)

=

119890 (119901 119902) + 119890 (119902 119903)

2

119890 (119901 119902) + 119890 (119902 119903) gt 1

0 119890 (119901 119902) + 119890 (119902 119903) le 1

(71)

But

1198652(119901 119903) =

119890 (119901 119902) + 119890 (119902 119903)

2

=

((1 minus 119901 + 119902) 2 + (1 minus 119902 + 119903) 2)

2

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

((1 minus 119902 + 119901) 2 + (1 minus 119903 + 119902) 2)

2

119901 lt 119902 lt 119903

=

2 minus 119901 + 119903

4

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

2 minus 119903 + 119901

4

119901 lt 119903 lt 119903

(72)



2(119901 119903) = 119890(119901 119902) and 119890(119902 119903) equiv 0 119890(119901 119902) + 119890(119902 119903) le 1 we


1(119901 119903) hArr



119890 (119901 119902) = 1 minus 119889 (119901 119902) (73)


119889 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

=

1 +1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

119901 = 119902

0 119901 = 119902

(74)



119890 (119901 119902) and 119890 (119902 119903) and dis (119902) le 119890 (119901 119903) (75)



L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)



1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865


spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)


forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr


4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)



holds


120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)


119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)



120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










L

M

Q

P



inc (119860119863)

=

1 119886 + 119889 gt 1 119886 = 119889 119886 gt 05 119889 gt 05

0 119886 + 119889 le 1

inc (119861 119863)

=

1 119887 + 119889 gt 1 119887 = 119889 119887 gt 05 119889 gt 05

0 119887 + 119889 le 1

inc (119862119863)

=

1 119888 + 119889 gt 1 119888 = 119889 119888 gt 05 119889 gt 05

0 119888 + 119889 le 1

(52)



equiv 1 (53)


119886119887119888119889[119901(119886) and 119901(119887) and 119901(119888) and 1] equiv 1











forall119901 119902 isin [0 1]

dArr

119889 (119901 119901) = 0

119889 (119901 119902) = 119889 (119902 119901)

119889 (119901 119902) + 119889 (119902 119903) ge 119889 (119901 119903)

(55)


119889 (119901 119902) = 0 lArrrArr

119901 = 119902

(56)


372 The t-Norm





119891 (119901 119902) = 119875 cap 119876 =

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

119891 (119902 119901) = 119876 cap 119875 =

119902 + 119901

2

119902 + 119901 gt 1

0 119902 + 119901 le 1

(57)

therefore 119891(119901 119902) = 119891(119902 119901)

Associativity


119891 (119901 119902) =

119901 + 119902

2

119901 + 119902 gt 1 997904rArr

119891 (119901 119902) and 119903 =

119891 (119901 119902) + 119903

2

119891 (119901 119902) + 119903 gt 1

0 119891 (119901 119902) + 119903 le 1

=

119901 + 119902 + 2119903

4

119901 + 119902

2

+ 119903 gt 1

0

119901 + 119902

2

+ 119903 le 1

(58)


1198911(119901 119903) =

119901 + 119902 + 2119903

4

119901 + 119902 + 2119903 gt 2

0 119901 + 119902 + 2119903 le 2

(59)



(59)

1198912(119901 119903) =

119902 + 119903 + 2119901

4

119902 + 119903 + 2119901 gt 2

0 119902 + 119903 + 2119901 le 2

(60)


2(119901 119903) = 0 |


2(119901 119903)


119901 and 119903 =

119901 + 119903

2

119901 + 119903 gt 1

0 119901 + 119903 le 1

119902 and 119903 =

119902 + 119903

2

119902 + 119903 gt 1

0 119902 + 119903 le 1

(61)



Neutrality Consider

1 and 119901 =

1 + 119901

2

1 + 119901 gt 1

0 1 + 119901 le 1

=

1 + 119901

2

119901 gt 0

0 119901 le 0

=

1 + 119901

2

119901 gt 0

0 119901 = 0

(62)


1 and 119901 =

1 + 119901

2

119901 isin (0 1)

119901 119901 = 0 119901 = 1

(63)

Absorption Consider

0 and 119901 =

119901

2

119901 gt 1

0 119901 le 1

(64)



forall119901 119902 isin [0 1]

dArr

119890 (119901 119901) = 1

119890 (119901 119902) = 119890 (119902 119901)

119890 (119901 119902) and 119890 (119902 119903) le 119890 (119901 119903)

(65)


119890 (119901 119902) = 119875 larrrarr

119876 =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(66)


Proof




119890 (119901 119902) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(67)

but

119890 (119902 119901) =

1 minus 119902 + 119901

2

119902 gt 119901

1 119902 = 119901

1 minus 119901 + 119902

2

119902 lt 119901

(68)

therefore 119890(119901 119902) equiv 119890(119902 119901)


From (66) let

1198651(119901 119903) = 119890 (119901 119903) =

1 minus 119901 + 119903

2

119901 gt 119903

1 119901 = 119903

1 minus 119903 + 119901

2

119901 lt 119903

(69)

119890 (119902 119903) =

1 minus 119902 + 119903

2

119902 gt 119903

1 119902 = 119903

1 minus 119903 + 119902

2

119902 lt 119903

(70)

then

1198652(119901 119903) = 119890 (119901 119902) and 119890 (119902 119903)

=

119890 (119901 119902) + 119890 (119902 119903)

2

119890 (119901 119902) + 119890 (119902 119903) gt 1

0 119890 (119901 119902) + 119890 (119902 119903) le 1

(71)

But

1198652(119901 119903) =

119890 (119901 119902) + 119890 (119902 119903)

2

=

((1 minus 119901 + 119902) 2 + (1 minus 119902 + 119903) 2)

2

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

((1 minus 119902 + 119901) 2 + (1 minus 119903 + 119902) 2)

2

119901 lt 119902 lt 119903

=

2 minus 119901 + 119903

4

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

2 minus 119903 + 119901

4

119901 lt 119903 lt 119903

(72)



2(119901 119903) = 119890(119901 119902) and 119890(119902 119903) equiv 0 119890(119901 119902) + 119890(119902 119903) le 1 we


1(119901 119903) hArr



119890 (119901 119902) = 1 minus 119889 (119901 119902) (73)


119889 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

=

1 +1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

119901 = 119902

0 119901 = 119902

(74)



119890 (119901 119902) and 119890 (119902 119903) and dis (119902) le 119890 (119901 119903) (75)



L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)



1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865


spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)


forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr


4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)



holds


120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)


119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)



120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119891 (119901 119902) = 119875 cap 119876 =

119901 + 119902

2

119901 + 119902 gt 1

0 119901 + 119902 le 1

119891 (119902 119901) = 119876 cap 119875 =

119902 + 119901

2

119902 + 119901 gt 1

0 119902 + 119901 le 1

(57)

therefore 119891(119901 119902) = 119891(119902 119901)

Associativity


119891 (119901 119902) =

119901 + 119902

2

119901 + 119902 gt 1 997904rArr

119891 (119901 119902) and 119903 =

119891 (119901 119902) + 119903

2

119891 (119901 119902) + 119903 gt 1

0 119891 (119901 119902) + 119903 le 1

=

119901 + 119902 + 2119903

4

119901 + 119902

2

+ 119903 gt 1

0

119901 + 119902

2

+ 119903 le 1

(58)


1198911(119901 119903) =

119901 + 119902 + 2119903

4

119901 + 119902 + 2119903 gt 2

0 119901 + 119902 + 2119903 le 2

(59)



(59)

1198912(119901 119903) =

119902 + 119903 + 2119901

4

119902 + 119903 + 2119901 gt 2

0 119902 + 119903 + 2119901 le 2

(60)


2(119901 119903) = 0 |


2(119901 119903)


119901 and 119903 =

119901 + 119903

2

119901 + 119903 gt 1

0 119901 + 119903 le 1

119902 and 119903 =

119902 + 119903

2

119902 + 119903 gt 1

0 119902 + 119903 le 1

(61)



Neutrality Consider

1 and 119901 =

1 + 119901

2

1 + 119901 gt 1

0 1 + 119901 le 1

=

1 + 119901

2

119901 gt 0

0 119901 le 0

=

1 + 119901

2

119901 gt 0

0 119901 = 0

(62)


1 and 119901 =

1 + 119901

2

119901 isin (0 1)

119901 119901 = 0 119901 = 1

(63)

Absorption Consider

0 and 119901 =

119901

2

119901 gt 1

0 119901 le 1

(64)



forall119901 119902 isin [0 1]

dArr

119890 (119901 119901) = 1

119890 (119901 119902) = 119890 (119902 119901)

119890 (119901 119902) and 119890 (119902 119903) le 119890 (119901 119903)

(65)


119890 (119901 119902) = 119875 larrrarr

119876 =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(66)


Proof




119890 (119901 119902) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(67)

but

119890 (119902 119901) =

1 minus 119902 + 119901

2

119902 gt 119901

1 119902 = 119901

1 minus 119901 + 119902

2

119902 lt 119901

(68)

therefore 119890(119901 119902) equiv 119890(119902 119901)


From (66) let

1198651(119901 119903) = 119890 (119901 119903) =

1 minus 119901 + 119903

2

119901 gt 119903

1 119901 = 119903

1 minus 119903 + 119901

2

119901 lt 119903

(69)

119890 (119902 119903) =

1 minus 119902 + 119903

2

119902 gt 119903

1 119902 = 119903

1 minus 119903 + 119902

2

119902 lt 119903

(70)

then

1198652(119901 119903) = 119890 (119901 119902) and 119890 (119902 119903)

=

119890 (119901 119902) + 119890 (119902 119903)

2

119890 (119901 119902) + 119890 (119902 119903) gt 1

0 119890 (119901 119902) + 119890 (119902 119903) le 1

(71)

But

1198652(119901 119903) =

119890 (119901 119902) + 119890 (119902 119903)

2

=

((1 minus 119901 + 119902) 2 + (1 minus 119902 + 119903) 2)

2

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

((1 minus 119902 + 119901) 2 + (1 minus 119903 + 119902) 2)

2

119901 lt 119902 lt 119903

=

2 minus 119901 + 119903

4

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

2 minus 119903 + 119901

4

119901 lt 119903 lt 119903

(72)



2(119901 119903) = 119890(119901 119902) and 119890(119902 119903) equiv 0 119890(119901 119902) + 119890(119902 119903) le 1 we


1(119901 119903) hArr



119890 (119901 119902) = 1 minus 119889 (119901 119902) (73)


119889 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

=

1 +1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

119901 = 119902

0 119901 = 119902

(74)



119890 (119901 119902) and 119890 (119902 119903) and dis (119902) le 119890 (119901 119903) (75)



L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)



1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865


spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)


forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr


4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)



holds


120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)


119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)



120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119890 (119901 119902) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(67)

but

119890 (119902 119901) =

1 minus 119902 + 119901

2

119902 gt 119901

1 119902 = 119901

1 minus 119901 + 119902

2

119902 lt 119901

(68)

therefore 119890(119901 119902) equiv 119890(119902 119901)


From (66) let

1198651(119901 119903) = 119890 (119901 119903) =

1 minus 119901 + 119903

2

119901 gt 119903

1 119901 = 119903

1 minus 119903 + 119901

2

119901 lt 119903

(69)

119890 (119902 119903) =

1 minus 119902 + 119903

2

119902 gt 119903

1 119902 = 119903

1 minus 119903 + 119902

2

119902 lt 119903

(70)

then

1198652(119901 119903) = 119890 (119901 119902) and 119890 (119902 119903)

=

119890 (119901 119902) + 119890 (119902 119903)

2

119890 (119901 119902) + 119890 (119902 119903) gt 1

0 119890 (119901 119902) + 119890 (119902 119903) le 1

(71)

But

1198652(119901 119903) =

119890 (119901 119902) + 119890 (119902 119903)

2

=

((1 minus 119901 + 119902) 2 + (1 minus 119902 + 119903) 2)

2

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

((1 minus 119902 + 119901) 2 + (1 minus 119903 + 119902) 2)

2

119901 lt 119902 lt 119903

=

2 minus 119901 + 119903

4

119901 gt 119902 gt 119903

1 119901 = 119902 = 119903

2 minus 119903 + 119901

4

119901 lt 119903 lt 119903

(72)



2(119901 119903) = 119890(119901 119902) and 119890(119902 119903) equiv 0 119890(119901 119902) + 119890(119902 119903) le 1 we


1(119901 119903) hArr



119890 (119901 119902) = 1 minus 119889 (119901 119902) (73)


119889 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

=

1 +1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

119901 = 119902

0 119901 = 119902

(74)



119890 (119901 119902) and 119890 (119902 119903) and dis (119902) le 119890 (119901 119903) (75)



L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)



1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865


spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)


forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr


4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)



holds


120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)


119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)



120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










L L998400

QP

MP998400

Q

P

(a)

M

L1 L2 L3

(b)



1198652(119901 119903) and dis (119902) le 119865

1(119901 119903) (76)

where 1198652(119901 119903) 119865


spondingly But

forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119903 + 119901) 4 + dis (119902))2

2 minus 119903 + 119901

4

+ dis (119902) gt 1

0

2 minus 119903 + 119901

4

+ dis (119902) le 1

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr

1198652(119901 119903) and dis (119902)

=

((2 minus 119901 + 119903) 4 + dis (119902))2

2 minus 119901 + 119903

4

+ dis (119902) gt 1

0

2 minus 119901 + 119903

4

+ dis (119902) le 1

(77)


forall119901 119902 119903 | 119901 lt 119902 lt 119903 997904rArr


4

forall119901 119902 119903 | 119901 gt 119902 gt 119903 997904rArr


4

(78)

that is we have

dis (119902) cong

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

0 119903 = 119901

(79)



holds


120575 (119901 119901) = 0

120575 (119901 119902) = 120575 (119902 119901)

120575 (119901 119902) or 120575 (119902 119903) or 119904 (119902) ge 120575 (119901 119903)

(80)


119890 (119901 119902) = 1 minus 120575 (119901 119902)

dis (119902) = 1 minus 119904 (119902) (81)



120575 (119901 119902) =

1 + 119901 minus 119902

2

119901 gt 119902

0 119901 = 119902

1 + 119902 minus 119901

2

119901 lt 119902

120575 (119902 119903) =

1 + 119902 minus 119903

2

119902 gt 119903

0 119902 = 119903

1 + 119903 minus 119902

2

119902 lt 119903

(82)


Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Given (82)

120575 (119901 119902) or 120575 (119902 119903) =

((1 + 119901 minus 119902) 2 + (1 + 119902 minus 119903) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

((1 + 119902 minus 119901) 2 + (1 + 119903 minus 119902) 2)

2

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 + 119901 minus 119903

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 gt 119902 gt 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

2 + 119903 minus 119901

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 lt 119902 lt 119903

=

2 +1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

120575 (119901 119902) + 120575 (119902 119903) lt 1 119901 = 119902 = 119903

1 120575 (119901 119902) + 120575 (119902 119903) ge 1

0 119901 = 119902 = 119903

(83)

but

120575 (119901 119903) =

1 + 119901 minus 119903

2

119901 gt 119903

0 119901 = 119903

1 + 119903 minus 119901

2

119901 lt 119903

(84)


120575 (119901 119902) or 120575 (119902 119903) le 120575 (119901 119903) (85)


119904 (119902) = 1 minus dis (119902) =

2 minus1003816100381610038161003816119901 minus 119903

1003816100381610038161003816

4

119903 = 119901

1 119903 = 119901

(86)


holds




120575 (119875 119876) fl inf 119889 (119901 119902) | 119901 isin 119875 119902 isin 119876 (87)

119904 (119875) fl sup 119889 (119901 119902) | 119901 119902 isin 119875 (88)


120575 (119871119872) fl inf 119889 (1198971015840 1198981015840) | 119897 isin 119871 119898 isin 119872 (89)

119904 (119871) fl sup 119889 (1198971015840 1198981015840) | 119897 119898 isin 119871 (90)




L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










L

Q

P

(a)

L

Q

P

(b)

L

Q

P

(c)


L

L998400

QP













119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871) (91)




119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119890 (119875 119876) [119871] equiv 119890 (119875 119876) and 119904 (119871)

=

119890 (119875 119876) + 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

0 119890 (119875 119876) + 119904 (119871) le 1

(92)

but from (66)

119890 (119875 119876) =

1 minus 119901 + 119902

2

119901 gt 119902

1 119901 = 119902

1 minus 119902 + 119901

2

119901 lt 119902

(93)




119890 (119875 119876) [119871]

=

1 minus 119901 + 119902 + 2 times 119904 (119871)

4

1 minus 119901 + 119902

2

+ 119904 (119871) gt 1

0

1 minus1003816100381610038161003816119901 minus 119902

1003816100381610038161003816

2

+ 119904 (119871) le 1

1 + 119904 (119871)

2

119901 = 119902

1 minus 119902 + 119901 + 2 times 119904 (119871)

4

1 minus 119902 + 119901

2

+ 119904 (119871) gt 1

(94)












(95)


inf119875119876

sup119871

[119877inc (119875 119871) and 119877inc (119876 119871)] (96)

inf119875119876119871119872



(97)



inf119875119876119871119872

[not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]]

and [119877inc (119875 119871) and 119877inc (119876 119871)]


(98)


inf119875119876119871119872



and 119877inc (119876119872)] 997888rarr 119890 (119871119872)

(99)


not119890 (119875 119876) [119871]

=

2 minus 119890 (119875 119876) minus 119904 (119871)

2

119890 (119875 119876) + 119904 (119871) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1

not119890 (119875 119876) [119872]

=

2 minus 119890 (119875 119876) minus 119904 (119872)

2

119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119872) le 1

(100)



not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










not119890 (119875 119876) [119871] and not119890 (119875 119876) [119872]

=


4


2

gt 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

0


2

le 1 119890 (119875 119876) + 119904 (119871) gt 1 119890 (119875 119876) + 119904 (119872) gt 1

1 119890 (119875 119876) + 119904 (119871) le 1 119890 (119875 119876) + 119904 (119872) le 1

(101)


inf119875119876119871119872


and 1 997888rarr 119890 (119871119872)

(102)



119894 119880 ⟩ 119864

119894isin



119864 119880 rarr [0 1] where 119880 = 0 1 2 10 is



119894]| forall119894 isin

[0Card119880] where

= int

119880

120583119864 (119906)

119906

(103)


119864(119906119894)119906119894| forall119894 isin


120583 (119906119894) = 1 minus

1

Card119880 minus 1


1003816100381610038161003816


(104)


119895 119880 ⟩




119863 119880 rarr [0 1]



= int

119880

120583119863 (119906)

119906

(105)



procedure

120583 (119906119895) = 1 minus

1

Card119880 minus 1

times

100381610038161003816100381610038161003816100381610038161003816


119863119895

075

]

100381610038161003816100381610038161003816100381610038161003816


(106)


119896 119881 ⟩




119878 119881 rarr [0 1] where 119881 = 0 1 2 10 is



119896]| forall119896 isin

[0Card119881] where

= int

119881

120583119904 (V)

V (107)


procedure

120583 (V119896) = 1 minus

1

Card119881 minus 1


1003816100381610038161003816


(108)



Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 3


119896isin 119881



119875 = ldquoIF (119878 is 1198751)AND (119863 is 1198752)




Cons 119890 is 1198761015840(110)



119877 (1198601 (119904 119889) 1198602 (

119890)) = [1198751 cap 1198752 times 119880] 997888rarr

119881 times 119876 = int

119880times119881

1205831198751 (119906)

(119906 V)and

1205831198752 (119906)

(119906 V)997888rarr

int

119880times119881

120583119876 (

V)(119906 V)

= int

119880times119881

([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))(119906 V)

(111)


[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V) =

1 minus [1205831198751 (119906) and 1205831198752 (

119906)] + 120583119876 (V)

2

[1205831198751 (119906) and 1205831198752 (

119906)] gt 120583119876 (V)

1 [1205831198751 (119906) and 1205831198752 (

119906)] le 120583119876 (V)

(112)

where [1205831198751(119906) and 120583

1198752(119906)] is min[120583

1198751(119906) 1205831198752(119906)] It is well


1015840cap11987521015840



119877(1198601(119904 119889) 119860

2(119890)) of type (111)

119877 (1198602 (119890)) = 1198751

1015840cap 11987521015840∘ 119877 (119860

1 (119904 119889) 1198602 (

119890)) = int

119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)]

119906

∘ int

119880times119881

[1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr

120583119876 (

V)(119906 V)

= int

119881

⋃

119906isin119880

[12058311987511015840 (119906) and 1205831198752

1015840 (119906)] and ([1205831198751 (119906) and 1205831198752 (

119906)] 997888rarr 120583119876 (

V))V

(113)



119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877 (1198601 (119904 119889) 1198602 (

119890))


= int

119880times119881

(120583119875 (119906) 997888rarr 120583

119876 (V))

and

((1 minus 120583119875 (119906)) 997888rarr (1 minus 120583

119876 (V)))

(119906 V)

(114)

where 119875 = 1198751 cap 1198752 and

119877 (1198601 (119904 119889) 1198602 (

119890))

= (120583119875 (119906) 997888rarr 120583

119876 (V))


119876 (V)))

=

1 minus 120583119875 (119906) + 120583119876 (

V)2

120583119875 (119906) gt 120583119876 (

V)

1 120583119875 (119906) = 120583119876 (

V) 1 minus 120583119876 (

V) + 120583119875 (119906)2

120583119875 (119906) lt 120583119876 (

V)

(115)











0

+

09

1

+

08

2

+

07

3

+

06

4

+

05

5

+

04

6

+

03

7

+

02

8

+

01

9

+

0

10

(118)


From (115) we have 119877(1198601(119904 119889) 119860

2(119890)) from Table 4

Table 4

1 09 08 07 06 05 04 03 02 01 01 1 045 04 035 03 025 02 015 01 005 009 045 1 045 04 035 03 025 02 015 01 00508 04 045 1 045 04 035 03 025 02 015 0107 035 04 045 1 045 04 035 03 025 02 01506 03 035 04 045 1 045 04 035 03 025 0205 025 03 035 04 045 1 045 04 035 03 02504 02 025 03 035 04 045 1 045 04 035 0303 015 02 025 03 035 04 045 1 045 04 03502 01 015 02 025 03 035 04 045 1 045 0401 005 01 015 02 025 03 035 04 045 1 0450 0 005 01 015 02 025 03 035 04 045 01



=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

09

5

+

1

6

+

09

7

+

08

8

+

07

9

+

06

10


=

07

0

+

08

1

+

09

2

+

1

3

+

09

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(119)

It is apparent that

119877 (1198601(1199041015840 1198891015840)) = 120583

119864 (119906) and 120583119863 (

119906)

=

04

0

+

05

1

+

06

2

+

07

3

+

08

4

+

08

5

+

07

6

+

06

7

+

05

8

+

04

9

+

03

10

(120)


1(1199041015840 1198891015840)) and 119877(119860

1(119904 119889) 119860

2(119890)) from Table 4

119877(1198602(1198901015840)) = 119877(119860

1(1199041015840 1198891015840)) ∘ 119877(119860

1(119904 119889) 119860


following 119877(1198602(1198901015840)) = 040 + 051+ 062+ 073+ 084+



119877 (1198602(1198901015840)) = 08 (121)


119890 (119871119872) isin [05 06] (122)


5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 Conclusion


Competing Interests


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Fuzzy Logic for Incidence Geometrydownloads.hindawi.com/journals/tswj/2016/9057263.pdf · ] we will use the following axioms from [ ]. es e axioms formalize the behaviour