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ON MEASURES OF FUZZINESS AND FUZZYCOMPLEMENTSMASAHIKO HIGASHI & GEORGE J. KLIRa Department of Systems Science, School of Advanced Technology, StateUniversity of New York , Binghamton, New York, U.S.A.Published online: 02 Apr 2008.

To cite this article: MASAHIKO HIGASHI & GEORGE J. KLIR (1982) ON MEASURES OF FUZZINESS AND FUZZYCOMPLEMENTS, International Journal of General Systems, 8:3, 169-180, DOI: 10.1080/03081078208547446

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Int J. General Systems, 1982, Vol. 8, pp. 169-1800308-1079/82/0803-0169 S06.50/0

© Gordon and Breach Science Publishers Inc.Printed in Great Britain

ON MEASURES OF FUZZINESS AND FUZZY COMPLEMENTS!

MASAHIKO HIGASHIft and GEORGE J. KLIRDepartment of Systems Science, School of Advanced Technology, State University of New York,

Binghamton, New York, U.S.A.

(Received December 18, 1981; in final form February 11, 1982)

An axiomatic framework for formalizing the most general class of fuzzy complements is introduced in this paper. It isthen used for investigating a general class of measures of fuzziness based on the view that the degree of fuzziness of afuzzy set should characterize the lack of distinction between the set and its complement. It is shown that the variousmeasures of fuzziness described previously in the literature are special cases of this general class. A class of so-calleddistance-based measures of fuzziness is also introduced. These measures are described in terms of the notion of anaggregation function of differences between the membership grades characterizing a fuzzy set and those of itscomplement. It is shown that the class of distance-based measures is equal to the class of general measures offuzziness. It is also shown that, given a particular fuzzy complement, aggregation functions which differ only in amultiplication constant represent the same measure of fuzziness.

INDEX TERMS: Fuzzy set, fuzzy complement, measure of fuzziness, distance, distance-based measures, sharpness offuzzy set, continuous fuzzy complement, involutive fuzzy complement, equilibrium of a fuzzy complement, differencefunction, aggregation function.

1 INTRODUCTION

With its growing maturity, the theory of fuzzysets35 becomes increasingly important for systemsresearch.5' 10' 20> 24> 25 The significance of thetheory, as a tool for dealing with imprecision, inthe investigation of complex systems, particularlyhumanistic systems, was well envisioned anddescribed by Lotfi Zadeh almost ten years ago37:

Essentially, our contention is that the conventionalquantitative techniques of system analysis are intrinsicallyunsuited for dealing with humanistic systems or, for thatmatter, any system whose complexity is comparable to that ofhumanistic systems. The basis for this contention rests onwhat might be called the principle of incompatibility. Statedinformally, the essence of this principle is that as thecomplexity of a system increases, our ability to make preciseand yet significant statements about its behavior diminishesuntil a threshold is reached beyond which precision andsignificance (or relevance) becomes almost mutually exclusivecharacteristics^ It is in this sense that precise quantitativeanalyses of the behavior of humanistic systems are not likely

•fThis work was supported in part by the National ScienceFoundation under Grant No. ECS-8006590.

{Also Department of Biophysics, University of Kyoto,Japan.

§A corollary principle may be stated succinctly as, "Thecloser one looks at a real-world problem, the fuzzier becomesits solution."

to have much relevance to the real-world societal, political,economic, and other types of problems which involve humanseither as individuals or in groups.

One of the issues associated with thefoundations of the theory of fuzzy sets is thequestion of how to measure fuzziness. Severalapproaches to the characterization of meaningfulmeasures of the degree of fuzziness of fuzzy setshave been proposed: De Luca and Termini6-7>8

proposed the concept of fuzzy entropy influencedprimarily by the Shannon entropy1'28; Kaufmannsuggested an index of fuzziness as a normalizeddistance (Euclidean or Hamming) between thegiven fuzzy set and a crisp set which is closest toit in the sense of the used distance19; Pollatschekassociated the degree of fuzziness with the lack ofdescriptive detail27; and Yager proposed to viewthe degree of fuzziness of a fuzzy set in terms ofthe degree of distinction between the set and itscomplement in such a way that the smaller thedistinction, the higher the degree offuzziness.32'33

Although the concept of fuzzy entropy hasinfluenced most contributors interested in thevarious questions regarding the measure offuzziness,4' 6"8> 13- 21- 22- 31 the characterizationproposed by Yager is, in our opinion, intuitively

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170 M. HIGASHI AND G. J. KLIR

the most appealing. We accept thischaracterization as a basis within which thequestion of possible measures of fuzziness isdiscussed in this paper.

If the degree of fuzziness of a fuzzy set isrelated to the lack of distinction between the setand its complement, then it is obvious that themeasure of fuzziness depends on the operator ofcomplement involved. It is well known that thecomplement operator is not unique for fuzzysets. 3- 10-12-14"18- »• 26- 29- 30- « • It is thusnecessary to discuss the measure of fuzziness inthe context of possible complement operators.

Accepting the intuitively appealing and generalview that the measure of fuzziness should in factmeasure the lack of distinction between a givenset and its complement, the aim of this paper isto clarify the relationship between the class ofpossible measures of fuzziness and the class ofpossible complement operators for fuzzy sets.

2 FUZZY COMPLEMENTS

Let U denote a crisp set which is taken as auniverse of discourse. Then, a fuzzy subset A of Uis defined by a function

where mA(u) characterisizes the grade ofmembership of u in /I.1 0-3 5

A complement of fuzzy set A is specified by afunction

which to each value mA{u) assigns a value ofc(mA(u)) and which must satisfy at least thefollowing two requirements (axioms):

(cl) c(0)=l and c(l) = 0, i.e., c collapses intoordinary complement for crisp sets;

(c2) if xl<x2, then c(x1)^c(x2), i.e., c ismonotonic decreasing.

All functions which satisfy (cl) and c2) formthe most general class of fuzzy complements. It israther obvious that the exclusion or weakening ofeither of these requirements would add to thisclass some functions totally unacceptable ascomplements. Indeed, requirements (cl) and (c2)are embedded in each of the various axiomatic

frameworks for fuzzy complements described inthe literature.3-10-12'17-23 Let these tworequirements be called the axiomatic skeleton forfuzzy complements.

In some instances, it is desirable to considervarious additional requirements for fuzzycomplements. Each of them reduces the generalclass of fuzzy complements to a special subclass.Two of the most desirable requirements, whichare usually listed among axioms of fuzzycomplements, are:

(c3) c is a continuous function;(c4) c is involutive, i.e., c(c(x))=x for all xe [0 ,1 ] .

It is well known23 that (c4) implies (c3), i.e.,every complement which is involutive is alsocontinuous. Hence, the set of all involutivecomplements is a subset of the set of allcontinuous complements which, in turn, is asubset of the set of all general complements.

An example of a class of fuzzy complementswhich satisfy only the axiomatic skeleton are thethreshold-type complements33 defined by

c(x) =1 for x g a

0 for x > a,

where ae [0 ,1 ] ,An example of a continuous (but not

involutive) fuzzy complement is the function

c(x)—i(l+cos7rx).

An example of a class of involutive fuzzycomplements is the Sugeno class10-29

Another example is the class

cw(x)=(l-xw)1^1, we(0,co),

which is meaningful in the context of the class offuzzy operators proposed by Yager34; this classwill be referred to as the Yager class of fuzzycomplements.

Each fuzzy complement c implies a function Con the set SPiJJ) of all fuzzy subsets of U,

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FUZZINESS AND FUZZY COMPLEMENTS 171

such that, for each Ae0>(U), C(A) = B if c{mA(u))=mB(u) for all ueU. Given a particular fuzzycomplement c, function C may conveniently beused as a global operator representing c.

In order to be able to investigate the relationbetween fuzzy sets and their complements, somerelevant properties of fuzzy complements must beintroduced first.

DEFINITION 1 Let c be a fuzzy complement.Then, any value of x for which c(x) = x will becalled an equilibrium of c.

LEMMA 1 Every fuzzy complement has at mostone equilibrium.

Proof Let c be an arbitrary fuzzycomplement. An equilibrium of c is a solution ofthe equation c(x)—x=0, xe[0 ,1] . We can provethe Lemma by demonstrating that any equationc(x)—x = d, where d is a real constant, has atmost one solution.f

Assume that xx,x2 are two distinct solutions ofthe equation c(x)—x = d such that xt<x2. Then,c(xi)—Xi=d and c(x2)—x2 = d; hence,

(1)

Since c is monotonic decreasing, c(x1)^x(x2) and,due to Xj < x 2 ,

This inequality contradicts to (1). Hence, theequation has at most one solution. Q.E.D.

LEMMA 2 Assume that a given fuzzy complementc has an equilibrium ec which must be unique(Lemma 1). Then,

and

x^c(x) iff x^

iff

Proof Assume x<ec, x = ec and x>ec,respectively. Then, c(x) = c(ec) c(x)=c(ec) andc(x)gc(ec), respectively, since c is monotonicdecreasing. From c(ec)=ec, we get c(x)^>ec, c(x)= ec and c(x)^ec, respectively, and due to theassumptions, we obtain c(x)>x, c(x)=x and

c(x)<x, respectively. Hence, x^ec implies c(x)^xand x ^ e c implies c(x)^x. The inverseimplications follow immediately. Q.E.D.

LEMMA 3 If c is a continuous fuzzy complement,then c has a unique equilibrium.

(Proof Given a fuzzy complement c, itsequilibrium ec is the solution of the equation c(x)—x = 0. This is a special case of a more generalequation c(x)—x = a, where ae[—1,1] is aconstant. By requirement (cl), c(0) —0=1 and c(l)— 1 = — 1. Since c is a continuous complement, itfollows from the intermediate value theorem forcontinuous functions2 that for each ae [— 1,1]there exists at least one x such that c(x)—x = a.fThe uniqueness of the solution is demonstratedin the proof of Lemma 1. Q.E.D.

For example, the equilibria for the individualfuzzy complements cx of the Sugeno class havethe following values:

e c =i /yi+; .- i for ;.=1/2 for A=0

For the complements cw of the Yager class, theequilibria are

eCw=(l/2)""'.

DEFINITION 2 Given a fuzzy complement c and areal number xe [0 ,1 ] , any real number dxe[0,1]such that

c(dx)-dx=x-c(x) (2)

is called a dual point of x with respect to c.

It follows directly from the proof of Lemma 1that Eq. (2) has at most one solution for theunknown dx (given c and x). Hence, there is atmost one dual point for each particular c and x.Moreover, it follows from the proof of Lemma 3that a dual point dx exists for each xe [0 ,1 ] whenc is a continuous complement.

LEMMA 4 If c has an equilibrium ec, then

fThis more genera! proposition is utilized later in thispaper.

fThis more general proposition is utilized later in thispaper.

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172 M. HIGASHI AND G. J. KLIR

Proof If x = ec, then x—c(x)=O by Definition1. To satisfy Eq. (2), we have to take dx= ec. Q.E.D.

LEMMA 5 For each xe[0,1], dx = c(x) iff c(c(x))= x.

Proof Let dx=c(x). Then, c(c(x))-c(x)=x— c(x); hence, c(c(x))=x. Let c(c(x))=x. Then,c(dx)-dx = c(c{x))-c(x); hence, dx=c(x). Q.E.D.

Lemma 5 means that the dual point coincideswith the complement point whenever thecomplement is involutive. If the complement isnot involutive, the dual point either does notexist or it does not coincide with complementpoint.

3 MEASURES OF FUZZINESS

Let 0(11) denote the set of all fuzzy subsets of U.Then, a measure of fuzziness is specified by afunction

which for each fuzzy subset A of U assigns avalue f(A). In order to accept f(A) as ameaningful characterization of the degree offuzziness of A,/must satisfy certain requirements.Although there is some flexibility in choosingthese requirements, they must fully capture themeaning of an accepted overall characterizationof the concept "degree of fuzziness." Moreover,they should be as general as possible in the sensethat they define the largest class of measuresacceptable by the overall characterization.

As mentioned in Section 1, we accept asthe intuitively most appealing overallcharacterization of the measure of fuzziness thatone according to which the degree of fuzziness ofa fuzzy set is viewed as the lack of distinctionbetween the set and its complement. Acceptingthis overall characterization, which is welljustified by Yager,32'33 the correspondingrequirements imposed upon / should be generalenough to encompass the whole class of fuzzycomplements.

In the spirit of the previous discussion, we cancapture the notion of the degree of distinctionbetween a fuzzy set and its complement byintroducing such a relation "being sharper than"

on the set &(U) that the distinction is greaterwhen the set is sharper.

DEFINITION 3 Let A, BeS?{U) and let

\mju) - c(mA{u))\ 1 \mB{n) - c{mB(u))\

for all u 6 U, where c is a fuzzy complement. ThenA is said to be sharper than B; formally, A<B.

Since the measure of fuzziness is supposed tocapture the lack of distinction between a fuzzyset and its complement, it is totally unacceptableto allow f(A)>f(B) if A<B. This leads to thefollowing requirements (axioms) for the measureof fuzziness/:

( / ) f(A)=0 iff A is a crisp set;(/2) it A<B, then f(A)^f(B);(/3) when c has an equilibrium ec,f(A) attains

its maximum iff mA(u) = ec for all u e U.

Requirement (/I), which simply states that acrisp set has no degree of fuzziness, is embeddedin each of the axiomatic frameworks for fuzzymeasures described in the literature.6' 9l 10> 13> 21>

22, 31, 32

Requirement (/2) is also common, as far as itsform is concerned, in the axiomatic frameworkspresented in the literature, but it is based on ageneralized concept of sharpness. The aim of thisgeneralization is to make the concept ofsharpness acceptable for the whole class ofgeneral fuzzy complements. It is clear that thecondition

\mA(u) - c(mA(u))\ ^ |mB(u) - c(mD(u))\

can be reformulated as

mA(u)^mB(u) or mA(u)-c(mA(u))

^c(mB{u))-mB{u) when mB(u)^c{)nB{u)),

and (a)

mJii)^mB(u) or mA{u)-c{mA{u))

^c(mB(ii))-mB(n) when mB(u)^c(mB(u)).

When the class of complements is restricted, weobtain the following special cases of the generalnotion of sharpness, derived directly from thisalternative formulation of sharpness:

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FUZZINESS AND FUZZY COMPLEMENTS

1. Continuous complements. A<B iff

mA(u)^mB(u) or mA(u)^m^du) when mB(u)<Lec

and (b)

mA(u) ̂ »iB(») or mA(u) ̂ JHBCK) when mB(u) k ec

for all ueU (by Lemmas 2 and 3).

2. Involutive complements. A<,B iff

HI^(H)^J«B(U) or /?JA(W)^C(HIB(U)) when »iB (w)^ec

and

HI^H)^»IB(U) or mA(ii)^c{uB{u)) when niB(u) §: <?c

for all ueU (by Lemma 5).

173

In case of the continuous complements, (B) isequivalent to

mA{u)^mB{u) when mB(u)g,ec

mA(u)^mB(u) when mB{u)^ec.

(Q

When the special complement c(x)=l— x is used(C) becomes equivalent to (A).

When definitions (A), (B), (C) are comparedwith (c), (a), (b), respectively, it is clear that thenotion of sharpness defined in this paper is ageneralization of that defined previously in theliterature. If A is a sharpened version of B in thesense of (B), then A<B. Therefore, (/2) impliesthe following requirement: 6- 9- 10- 13- " • 22- 31

3. Special complement c(x)= 1 — x.

mA(n)^mB{n) or mA{u)^l —mB{u) when

and

mA(u)^mB(u) or mA(u)£l —mB(u) when

iff

u)^ 1/2

(c)

«)^ 1/2

(/2)' if A is a sharpened version of B in the sense

for all H E U.The only notion of sharpness presented by the

various authors in the literature6- 9> I0> 13> 21> 22>

31 is based on the operation of contrastintensification proposed by Zadeh.36 It is definedas follows: A is said to be a sharpened version ofBiff

mA(u)f^mB(u) when mB(u)^ 1/2

and (A)

mA(u)^mB{u) when mB(u)^l/2.

This definition is obviously applicable only to thespecial complement c(x) = l— x, but can beextended for the class of all complements in thefollowing way:

mA{u)£mB(u) when mB(u)^c(mB(u))

and (B)

Requirement (/3), is again a generalization ofthe usual requirement of maximal value of f(A)—f(A) is maximum iff mA(u) = 1/2 for all u e U6' 9i

10, 13, 2i, 22, 31 ^ j ^ j s tailored to the specialfuzzy complement c(x)=l— x. Observe that therequirement is vacuous when the used fuzzycomplement does not have an equilibrium.

4 DISTANCES BETWEEN FUZZY SETSAND COMPLEMENTS

In order to characterize the distinction betweenfuzzy sets and their complements, possibledistance measures are investigated in this section.

Assume first that the universe of discourse U isfinite and let C/ = {»1,u2,...,wn}. Given acomplement c and a fuzzy set A e &\u\ let

be a function such that

<5C, ^(".) = | ' " A ( " I ) - c(mA(

and let

/de:0'(U)-+[O,lY

be a function such that

(3)

mA(u)^.mB(u) when mB(ii)^c(mB{u))

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174 M. HIGASHI AND G. J. KLIR

the superscript / is used here to indicate that thefunction is defined for a finite universe U.

In order to characterize each particular fuzzycomplement, let

be a function such that

5c(x)=\x-c(x)\

and let

Ac = {<5c(x)|xe[0,l]};

clearly, Acc[0,1]. Let 8c be called a differencefunction associated with fuzzy complement c.

LEMMA 6 fdc(^(U)) = Anc when U is finite.

Proof Consider any Ae^P{U). SincemA{iii)e[0,1] for any w.el/, 5c>/1(u,-)eAe for eachi/,e[/ due to the definition of 5C A and,consequently, hlc(A)e A"c. Hence, fdc{0>(U))^A"c.

Consider now any (5C(-Xi), <5c(x2),..., <5c(xn)) 6 A".Then we can find a fuzzy set Ae£?(U) such thatfdc(A)=(^c{xi), <5c(x2),...,<5c(xn)), namely, A isdefined by mA{n,)=Xi for all u{eU. Hence,

Q.E.D.

Given a fuzzy complement C, it is reasonable toview the distance between A and C(A) as somekind of aggregation function of the localdifferences dCiA(u^ in membership grades of A andC(A) for each element of U.

DEFINITION 4 Given a fuzzy complement c andassuming that U is finite, let

$c,A' Sc IS viewed as a measure of distance

between A and C(A). Let DCt9(A, C(A)) denote thisdistance. Then

DcJA,C(A))=gc(fdc(A)).

Assume now that U is infinite. The previousdefinitions can easily be modified as follows: (3) isreplaced by

» ) | , " e U>

fdc becomes defined as

such that

where 8CyA is a difference function of A withrespect to c.

LEMMA 7 dc(0>(U))=ACU when U is infinite.

Proof A trivial modification of the proof ofLemma 6.

Lemma 7 means that the consideration of anarbitrary oceA^ is equivalent to the considerationof an arbitrary difference function SCtAedc(^(U)).

There is also a need to adjust the definition ofthe aggregation function for the case when U isinfinite.

DEFINITION 5 Given a fuzzy complement c andassuming that U is infinite, let

be a function which has the following properties:(gl) gc{aua2,...,a^Q for each {al,a2,...,an)eAn

c;(gl) gc is monotonic increasing with respect to its

every argument;(g3) gc{aua2,...,an) attains its maximum iff a~\

for all i;

(g4) if OeA,., then gc{aua2,...,an) attains itsminimum iff at=0 for all f.

Then gc is called a finite aggregation functionassociated with the fuzzy complement c.

When arguments of the finite aggregationfunction gc are values dCyA(ui), u,eU, of function

be a function which has the following properties:fel)' ^(cO^OforanyaeAf;

gc is monotonic increasing in the sense thatgc(oc)^gc(a') when a(ii)§a'(ii) for all ueU;gXa) attains its maximum iff a(w)=l for allueU;if OeA, then g(a) attains its minimum iffa(H) = 0for all ueU.

Then gc is called a continuous aggregation functionassociated with fuzzy complement c.

Observe that Definition 5 is a generalization ofDefinition 4, i.e., the notion of the finiteaggregation function can be viewed as a special

(g4)'

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FUZZINESS AND FUZZY COMPLEMENTS 175

case of the notion of the infinite aggregationfunction. Hence, when there is no need todistinguish between finite and infinite aggregationfunctions, we may call any function which ischaracterized by Definition 5 an aggregationfunction.

When the argument of the infinite aggregationfunction gc is represented by a difference function5C A, gc is again viewed as a measure of distance/and C(A).

Examples of the introduced concept of adistance between A and C(A) are the individualmembers of the Minkowski class of distances. If1/ = {H1,«2,- . . ,«„}, they are defined by

where p=l,2,... is a parameter whose valuescharacterize the individual types of distances (e.g.,Hamming distance for p = l, Euclidean distancefor p=2, upper bound distance for p = co). If U= [_a,b], then

De,plA,C(A)} =

Any distance DCtg[_A,C(Ay], AE&>(U), satisfiesthe inequalities

where N is an arbitrary crisp set in ^(U). Anormalized version Dc g of Dc g is defined by

Clearly,

5 DISTANCE-BASED MEASURES OFFUZZINESS

The concept of distance, as introduced inSection 4, represents a basis within which thedistinction between fuzzy sets and theircomplements can be characterized in a number ofalternative ways. Since the measure of fuzziness /

should characterize the lack of this distinctionand is required to be normalized, we have

(4)

for all As&{JJ). Let a measure of fuzziness whichsatisfies (4) be called a distance-based measure offuzziness.

We come now to the formulation of the majorresults of this paper: the relationship between thegeneral measures of fuzziness, which satisfyrequirements (/I) , (/2), (/3), and the distance-based measures.

THEOREM 1 All distance-based measures offuzziness fc g satisfy the conditions ( / I) , (/2) and

Proof

(/I) .M c J { ) \og(dc(A))=g{dc(N)),

where N is an arbitrary crispsubset of U,

^c/»(") = 1 for all tie UomA{u)=l or 0 for all ueU

oA is a crisp subset of U.

The third equivalence above is due to (g3)';the fourth equivalence is due to (cl) andthe proposition involved in the proof ofLemma 1.

(/2) Suppose A<.B, i.e.

Then, due to (g2)', we get

Hence, feJA)£feJB).

(/3) Suppose c has an equilibrium ec. Since c(ec)= ec, \ec-c(ec)\ = 0 and e,.e[0,1]. Thus,OeAc. Hence, due to (g4)', g{5c<A) attains itsminimum iff 5C>j4(u)=0, i.e., mA(u)=c(mA(u)),for all ueU. Therefore, fc,g{A) attains itsmaximum iff mA{u)=c(mA(u)) for allueU. Q.E.D.

Since (/2) implies {f2)',fcg also satisfies (/2)'.This is a generalized result of that by Yager32

which means that for the special complement c(x)= l—x and the special aggregation function

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176 M. HIGASHI AND G. J. KLIR

(Minowski type)

p-*(«) = f {«('

Let

= 1 -UPP

,fc,g satisfies (/2)'.

{ | ^ ( « ) - * [ 0 , l ] satisfies (/I), (/2) and(/3) for a particular c} and let

where

9t = {g\g: Ay->[0.1] satisfies fcl)', fe2)\ fe3)'and (g4)' for a particular c}.Then, Theorem 1 can be restated as

The next theorem indicates that the inverseproposition also holds.

THEOREM 2 3ic^!F c, that is, for any/eJ%, thereexists ge^/c such that

Proof Consider any fe£rc. Let

such that

f(A)=l-f(A).

Consider d~l ojt a correspondence from A" to[0,1], where d~l denotes the inverseof function dc; then we can state the follow-ing two propositions (proven later);

i) The correspondence d~l °f is a functionfrom Av

c to [0,1], i.e., df1 of: A?-»[0,1].ii) The function rff1"/: A?-»[0, 1] satisfies

fel)', fe2)', fe3)' and (g4)'. Hence, d;l°fe<3c. Letgo=dc

l °f; then it can be seen that for any

= 1 -f{d;\dc{A)))

T(A)T(N)

= 1-(1-/(/!))=/(/!),

where N is an arbitrary crisp subset of U.The third equality above holds due to the

proposition (ii) and the fifth holds due to (/I).Hence, f=fc,g , which concludes the proof.

Proof of (i)

Consider any aeAf and its image by thecorrespondence d~i °f, i.e.,

Here,

d-1{a)={A\Ae&(U) and dc{A)=a}.

Consider any AuA2ed~l(a); then

, i-e.,

\mA,(") -

for all net/. This implies At~<.A2 and A2<A1;hence, due to (/2), we get f{A1)^f{A2) and/(i42)^/"Mi), respectively. Thus, /(A)=/(/l2);hence /(/ii)=/(^2)- This concludes that anyAed~l{<x) corresponds to the same element of[0,1] byf, which implies that the image of ^"'(a)by f, i.e., the image of a by d~a °J, consists of asingle element of [0,1]. Therefore, thecorrespondence d~1 °f is a function from Au to[0,1].

Proof of (ii)

(gl)' Since dc{0>(U))=A¥, for any aeA? thereexist Ae0>(l)) such that dc(^)=a. Then,according to (i), d;1 of{a)=f(d-l{dc{A)))=/(/!). Since f(A)=l-f{A)^0, weconclude that d'1 °/(oc)^O.

(g2)' Consider auu2eA" such that ai(»)^a2(t()for all u eU. Since dJ&{U))=£g, for «j anda2 there exist /I1 ; /l2 such that

dc(A2) = a2. Then,

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FUZZINESS AND FUZZY COMPLEMENTS 177

\mAl(u) - c(mAl(u)\ ^ \mAi(u) - c(mAi(u))\

for all ueU.

Hence, due to (/2), we get

which implies f(A1)^f(A2). On the other hand,according to (i),

(d-l o/)(a2) =f(de- \dc{A2)) =f(A2).

Therefore, (Jf1 o/XcO^f1 °/)(<x2).

(g3)' For any aeAf, there exists Ae&iU) suchthat rfc(^) = a ; then according to (i) and dueto (/"I), we get

=7(4 = 1 -f(A) g 1,

where, due to (/I), the equality holds iff Ais a crisp subset of I/. And, according tothe proof of Lemma 1 and due to (cl), A isa crisp subset of U, i.e., mA(u) = l or 0 forall wet/, iff »»,,(«)—c(mx(n))= ± 1 , i.e.,|'n^(")—c('»x("))| = l f°r all wet/. Therefore,the equality holds in the previousinequality iff a(»)= 1 for all ueU.

(g4)' For any aeAf, there exists A e 0*(U) suchthat d~(A) = a; then, according to (i) anddue to (/3), we get {d;loj){a)=J{d; \dlA)))=J{A) = \ -f(A)*0, where,due to (/3), the equality holds iff, for allu e U, mA(u) = ec, i.e., a(u) = \mA(u)-c{mA(u)\= 0. Q.E.D.

THEOREM 3 ®<.=Jzrc.

Proof Follows immediately from Theorem 1and Theorem 2.

Based on Theorem 3, we can define a function

such that ec(g)=fCi9

fwhere ge<Sc and fc,ge3?c; it

ontothe

i9follows from the theorem that ec is anfunction and, consequently, it inducesfollowing equivalence relation ~ in 0c:(Vg,,gJeye)(gi*'gj iff ejig,) = ec(gj)). The associatedpartition <8cl&

j consists of the inverse image of theelements of/by ec, i.e., gj*< = {e;\f)\fs!Fc}.

Next question of our interest is how big classof aggregation functions (subset of yc)corresponds to a measure of fuzziness/6^, i.e.,what is the inverse image of/ by ec, e~i(f). Or,what is the partition of 2?c induced by ec. S?<./~,each of whose elements (equivalence classes)corresponds to each of distinct measures offuzziness in a one to one fashion.

Let the function ec defined above be called anexpression function and let ge2?c be called anexpression of fe^c; f is expressed by g if ec{g)—fcg=f- Then, e~l(f) represents the set ofall the expressions of/.

THEOREM 4 Supposefe&a i.e., ec(gf)=f.Then,

/c is an expression of

and g{oi) = (5)

Proof Assume geec ' ( /) , i.e., ec(g)=fc,g=f.Then, for all A e 0>(U),

f (A\=X s(dc(A))}c-s{ > g(dc(N))'

where N is an arbitrary crisp subset of U. Sincegf is an expression of/

Therefore

g(d(A))= g(d(A))

g(d(N)) gf{d(N))' ••'

g{d(A))=^^gf(d(A))(ovany

Since dc{0>(U)) = Av, we get

g(d(N))

where

*(«)=f7^»forany*eAt;>

g(d(N))

gf(d(N))

is a positive constant. Therefore, g belongs to theset on the right-hand side of (5).

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178 M. HIGASHI AND G. J. KLIR

Conversely, assume that g(<x)=kgj{a) for anyaeA17, where k is a positive constant. For any

), we get

Due to Theorem 3, we can conclude that

g(dc{N)) kg,(dc{N))

Since gf is an expression of/, i.e., ec(gf)=fc =/,/cJ/1) =/(-4) for any /I e0>{U).

Hence,

for any ,4 e£

i.e., ec{g)=fCt9=f,

which means that g is also an expression of/, i.e.,gee~'(/) . This concludes the proof. Q.E.D.

Let ~ denote an equivalence relation in tScdefined as follows:

Ofei, gj 6 0,)

for some fc>0 and all aeA17).

Theorem 4 indicates that e~l(f) = [g/\, where ^is an expression of / and [g/] denotes theequivalence class in (SJ~ which contains gf.

Since

we may conclude that

We can also define the classes

6 CONCLUSIONS

A general class 8F of measures of fuzziness isintroduced in this paper on the basis of the viewthat the degree of fuzziness of a fuzzy set shouldcharacterize the lack of distinction between theset and its complement. In order to describe themost general class of measures of fuzziness, basedon this view, the most general class of fuzzycomplements is introduced and investigated.

It is demonstrated that the class J5" of allgeneral measures of fuzziness is equal to the class3) of such measures of fuzziness in which the lackof distinction of the considered set and itscomplement is expressed in terms of some formof metric distance. This set equality, &=Q),suggests that every function

such that

where Dc g[A, C(A)] denotes a normalized metricdistance between fuzzy set A and its complementC(A), is acceptable as a measure of fuzziness and,conversely, every measure of fuzziness can beexpressed in terms of some function fcg and theassociated metric distance function Dcg.

The metric distance function Dcg is determinedby two choices: (i) a complement c, and (ii) anaggregation function g by which the individualdifferences

and

of all general measures of fuzziness and distance-based measures of fuzziness, respectively, where

(6=\c\c is a fuzzy complement}.

are aggregated for all elements of the universe ofdiscourse U. The complement may often be fixedby the area of application or otherconsiderations. Then, the determination of anacceptable measure of fuzziness is solely a matterof the selection of an aggregation function fromthe class (SC associated with the givencomplement c. It is shown that, given a particularcomplement, each equivalent class of aggregationfunctions whose elements differ from each other

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FUZZINESS AND FUZZY COMPLEMENTS 179

only in positive multiplication constant representsa measure of fuzziness different from thoserepresented by other equivalent classes ofaggregation functions.

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8. A. De Luca and S. Termini, "On the convergence ofentropy measures of a fuzzy set." Kybernetes, 6, No. 3,1977, pp. 219-227.

9. A. Di Nola and S. Sessa, "On the fuzziness measure andnegation in totally ordered lattices." Busefall, 8, Autumn1981, pp. 68-77.

10. D. Dubois and H. Prade, Fuzzy Sets and Systems: Theoryand Applications. Academic Press, New York, 1980.

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180 M. HIGASHI AND G. J. KLIR

Masahiko Higashi was born onAugust 11, 1954 in Kyoto, Japan.He received a B.S. degree in biologyand mathematics in 1978 and a M.S.degree in biophysics (mathematicalbiology) in 1980, both from theUniversity of Kyoto. After receivinghis M.S/ degree, he was admitted tothe D.S. program at the Universityof Kyoto. In Fall 1980, he joined thePh.D. program at the School of

Advanced Technology, State University of New York atBinghamton, where he is now a Ph.D. candidate in systemsscience. His main current interests are in systems theory andmethodology as a means for developing and integratingecological theories. He is a member of the Japan Society ofEcology, Japan Society of Biophysics, and Society forGeneral Systems Research.

George J. Klir is a Professor ofSystems Science and Chairman ofthe Department of Systems Science,School of "Advanced Technology,State University of New York atBinghamton. He received the M.S.degree in Electrical Engineering fromthe Prague Institute of Technologyin 1957, and the Ph.D. degree inComputer Science from theCzechoslovak Academy of Sciences

in 1964. He is also a graduate of the IBM Systems ResearchInstitute in New York.

Before joining the State University of New York, Dr. Klirhad been with the Computer Research Institute and CharlesUniversity in Prague, the University of Baghdad, theUniversity of California at Los Angeles, and FairleighDickinson University in New Jersey; he has also worked parttime for IBM and Bell Laboratories, and taught summercourses at the University of Colorado, Portland StateUniversity in Oregon and Rutgers University in New Jersey.During the academic year 1975-76, he was a Fellow at theNetherlands Institute for Advanced Studies in Wassenaar,Holland, and in 1980 he was a fellow of the Japan Society forthe Promotion of Science.

Dr. Klir's main research activities have been in generalsystems methodology, logic design and computer architecture,switching and automata theory, discrete mathematics and thephilosophy of science. He is the author of over eighty articlespublished in various professional journals and holds anumber of patents. He published twelve books, among themCybernetic Modelling (IlifTe, London, 1967), An Approachlo General Systems Theory (Van Nostrand, New York, 1969),Trends in General Systems Theory (John Wiley, New York,1972), and Methodology of Snitching Circuits (Van Nostrand,New York, 1972).

Since 1972, Dr. Klir has been Editor-in-Chief of theInternational Journal of General Systems. He is Editor ofthree book series and a member of Editorial Boards ofthirteen journals. He is a senior member of IEEE, and amember of AAAS, SGSR, SIAM and PSA (Philosophy ofScience Association). He is currently serving as President ofthe International Federation of Systems Research andPresident of the Society for General Systems Research.

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