STUDIES ON INTUTIONISTIC FUZZY INFORMATION MEASURE

Ist International Conference of ISITA on

Mathematical Modeling, Optimization and Information TechnologySSCET, Badhani, Pathankot, Punjab

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INFORMATION MEASURE

Surender SinghAssistant Prof.

School of Mathematics, Faculty of Sciences

Shri Mata Vaishno Devi University

Katra –182320 (J & K)

Email:surender1976@gmail.com16th -19th Jan., 2015

OUTLINE

INTRODUCTION AND PRELIMINARIES

SPECIAL T-NORM

INTUTIONISTIC FUZZY ENTROPY

EXAMPLES OF REF’s AND NEW MEASURE

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EXAMPLES OF REF’s AND NEW MEASURE

OF IFE

CONCLUDING REMARKS

1. INTRODUCTION AND PRELIMINARIES

Fuzziness, a feature of imperfect information, results from the lack of crisp distinction between the elements belonging and not belonging to a set (i.e. the boundaries of the set under consideration are not sharply defined).

The concept of fuzziness initiated by [Zadeh, 1965].

[De Luca and Termini, 1971] introduced some requirements which capture our intuitive comprehension of the degree of fuzziness in a fuzzy capture our intuitive comprehension of the degree of fuzziness in a fuzzy set and introduced concept of fuzzy entropy.

The term ‘Fuzzy entropy’ have been adopted due to an intrinsic similarity of equation to the one in the [Shannon, 1948] entropy.

Two functions measure fundamentally different types of uncertainty. Basically, the Shannon entropy measures the average uncertainty in bits associated with the prediction of outcomes in a random experiment.

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The idea of intutionistic fuzziness initiated by [Atanassov, 1986] and incidentally affected all fields of study where ever concept of fuzziness was used. [Vlachos et al., 2007] derived an extension of De Luca- Termini’s entropy for IFSs. In this paper, we study measure of fuzziness for intuitionistic fuzzy sets in light of restricted equivalence functions and T-norm operator.

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Definition 1.1.1[Bustince et al., 2006] Let cc ],1,0[]1,0[: is a fuzzy

negation iff:N1: 1)0( c and ,0)1( c

N2: yxifycxc ,)()( (monotonicity).

A fuzzy negation is strict, iff,N3: )(xc is continuous.

N4: yxifycxc ,)()( for all ].1,0[, yx

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N4: yxifycxc ,)()( for all ].1,0[, yx

A strict fuzzy negation is involutive, iff,N5: ),())(( xcxcc ].1,0[x

The strict fuzzy negations that are involutive are called strong negations.

Definition 1.1.2[Bustince et al., 2006] A function ]1,0[]1,0[: 2 REF is called a

restricted equivalence function, if it satisfies the following conditions:R1: ),(),( xyREFyxREF for all ];1,0[, yx

R2: 1),( yxREF iff ;yx

R3: 0),( yxREF iff 1x and 0y or 0x and ;1y

R4: ))(),((),( ycxcREFyxREF for all ],1,0[, yx c being a strong negation;

R5: For all ],1,0[,, zyx if ,zyx then ),(),( zxREFyxREF

and

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and .),(),( zxREFzyREF

It can be proved that R5 is equivalent to: for all x, y, z, t [0, 1], if ,tzyx then .),(),( txREFzyREF

1.2 Intutionistic fuzzy sets

Let X be an ordinary non-empty, finite set. Then an IFS over X is

characterized by two mappings ]1,0[: Xg and ].1,0[: Xh

For each )(, xgXx can be interpreted as the degree to which x enjoys some

property P. Alternately, )(xh is the degree to which x does not enjoy the property P. Here, g and h are the generalizations of characteristic function of conventional set theory. There is nothing intutionally fuzzy about set X. Rather, the fuzziness lies in the degree of compatibility and the degree of

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Rather, the fuzziness lies in the degree of compatibility and the degree of incompatibility of the element Xx with property P. Definition 1.2.1. An intutionistic fuzzy set A defined on a universe X is given by [Atanassov, 1986]:

},|))(),(,{( XxxhxgxA AA

where,

]1,0[: XgA and ]1,0[: XhA , with the condition ,1)()(0 xhxg AA for all Xx .The numbers )(xgA and )(xhA denotes the degree of membership and the degree of non membership of x to A, respectively. For all IFS A in X we call the intutionistic index of an element Xx in A the following expression:

).()(1)( xhxgx AAA

we consider )(xA as a hesitancy degree of x to A.

Evidently, for all .

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Evidently, 1)(0 xA for all Xx .1.2 Fuzzy Entropy

In fuzzy set theory, the entropy is measure of fuzziness which expresses the amount of average ambiguity /difficulty in taking a decision weather an element belong to a set or not. A measure of fuzziness H (A) of a fuzzy set A should have the at least the following four properties .

FE1 (Sharpness): H (A) is minimum if and only if A is crisp set, that is, A(X) =0 or 1 for all .Xx

FE2 (Maximality): H (A) is maximum if and only if A is most fuzzy set, that is, A(X) =0.5 for all .Xx

FE3 (Resolution): H (A)H(A*),where A* is the sharpened version of A.

FE4(Symmetry):H(A)=H(�),where � is the complement set of A , that is ,�(x)=1- A(x) for all .Xx

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[Ebanks, 1993] proposed one more axiom as essential condition for validity of a measure of fuzzy entropy

FE5 (Valuation): )()()()( BHAHBAHBAH .

[DeLuca and termini, 1971] introduced a measure of fuzziness analogous to the information theoretic entropy of [Shannon, 1948] as

n

iiAiAiAiA xxxx

nAH

1

))(1log())(1()(log)(1

)( .

2. Special T-Norm

The truth table of the classical binary conjunction ^ is given in Table

1. In many-valued logic we extend the classical binary conjunction to

the unit interval as a ]1,0[]1,0[ 2 mapping as follows:

Definition 2.1 [Schweizer and Sklar, 1960] A mapping ]1,0[]1,0[: 2 C is a

conjunction on the unit interval if it satisfies:

C1.Boundary conditions: C(0, 0) = C(0, 1) = C(1, 0) = 0 and C(1, 1) = 1,C1.Boundary conditions: C(0, 0) = C(0, 1) = C(1, 0) = 0 and C(1, 1) = 1,

Table 1: Truth table of the classical binary conjunction

p q qp

0 0 0

0 1 0

1 0 0

1 1 1

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C2. Monotonicity: )]1,0[),,(( 3 zyx )),(),(),(),(( yzCxzCandzyCzxCyx

Definition 2.2 [Schweizer and Sklar, 1960] A mapping ]1,0[]1,0[: 2 T is a

triangular norm

(t-norm for short) if for all 2]1,0[,, zyx it satisfies:

T1. Boundary condition: ,)1,( xxT T1. Boundary condition:

T2. Monotonicity: ),(),( zxTyxTzy ,

T3. Symmetry: ),(),( xyTyxT ,

T4. Associativity: )),,(()),(,( zyxTTzyTxT .

A t-norm T always satisfies .]1,0[),( xxxxT

If we put a restriction on T that ]1,0[1),( xxxT .

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Now this idea of this t-norm may further be extended as follows:

Definition 2.3 A mapping ]1,0[]1,0[]1,0[: nnRT is a special triangular

norm if for all

]1,0[,,]1,0[)...,,,(),...,,,(),...,,,( 212121 iiin

nnn zyxandzzzyyyxxx zyx it satisfies:

RT1. Boundary condition: ,)( x1x, RT

RT2. Monotonicity: ),(),( zxyxzy RR TT ,

RT3. Symmetry: ),(),( xyyx RR TT ,RT3. Symmetry: ),(),( xyyx RR TT ,

RT4. RT is permutationally symmetric,

RT5. Associativity: )),,(()),(,( zyxzyx RRRR TTTT ,

RT6. nRT ]1,0[),( x1xx ,

RT7. .]1,0[),(),(),( nRRR TTT xy00xyx

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3. Intutionistic Fuzzy EntropyLet ]}1,0[:|{]1,0[ XggX

g and ]}1,0[:|{]1,0[ XhhXh then a measure of

degree of intutionistic fuzziness is a non negative function ]1,0[]1,0[]1,0[: X

hXgd such that a functional can be regarded as an entropy in

the sense that it measures our uncertainty about the presence, absence or indeterminacy of some property P over X.

Some desirable properties of intutionistic fuzzy entropy:Intutionistic fuzziness of a set XA is characterized by two functions

]1,0[: Xg A and ]1,0[: XhA , with the condition ,1)()(0 xhxg AA for all

Xx . we have X

gAg ]1,0[ and XhAh ]1,0[ .

Let }...,,,{ 21 nxxxX , )...,,,( 21 ngggAg with nixgg iAi ...,,2,1),( and

)...,,,( 21 nhhhAh with nixhh iAi ...,,2,1),( .

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Here, following facts are clear:(i) XX

hXg ]1,0[]1,0[]1,0[ .

(ii) n

timesn

]1,0[]1,0[...]1,0[]1,0[ Ag and n

timesn

]1,0[]1,0[...]1,0[]1,0[ Ah .

Let ),( AA hgf , we say .]1,0[]1,0[ nnf

Now, define ),()...,,,,...,,,(),()( 2121 AAAA hghg R

termsn

n

termsn

nR ThhhgggTdfd

, (1)

for some function ]1,0[]1,0[]1,0[: XXRT (definition 2.3)

,

Now it is convenient to impose a lattice structure on XX ]1,0[]1,0[ , as

follows:Let XXff ]1,0[]1,0[, be such that ),( AA hgf and ),( AA hg f .we define

)})(),(.{min},)(),(.{(max),( xhxhxgxghhggff AAAAAAAA and

)})(),(.{max},)(),(.{(min),( xhxhxgxghhggff AAAAAAAA .

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From the above notions, a list of essential properties for measure of intutionistic fuzziness follows.P1 Sharpness: We have, 0)( fd i.e. 0),( AA hgRT iff. },1,0{)(),( XhXg AA i.e.

Ag and Ah are sharp.

P2 Maximality: ),(),()( AA hgRAA Thgdfd attains its maximum value only

when AA hg i.e. .AA hg

P3 Resolution: )()( fdfd iff ),(),( AAAA hgdhgd iff ),(),( AAAA hghg RR TT

if gg and hh for hg i.e. if gg and hh for hg .if AA gg and AA hh for AA hg i.e. if AA gg and AA hh for AA hg .

orif AA gg and AA hh for AA gh i.e. if AA gg and AA hh for AA gh

P4 Symmetry: ),(),( AAAA ghdhgd ie. )()( AAAA g,hh,g RR TT .P5 Valuation: )()()()( fdfdffdffd

i.e f and f can exchange intutionistic fuzziness at various points

without affecting the sum of the degrees of intutionistic fuzziness.

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Lemma 3.1 Let }...,,,{ 21 nxxxX . A measure of fuzziness ]1,0[]1,0[]1,0[: XXd

satisfies valuation property iff. there exists a map ]1,0[]1,0[]1,0[: such that

.]1,0[]1,0[))(),(()(

1

XXn

iiAiA fxhxgfd

(2)

Lemma 3.2 Let }...,,,{ 21 nxxxX and suppose that ]1,0[]1,0[]1,0[: XXd is

given by (2) for some ]1,0[]1,0[]1,0[: . Then

(a) d satisfies P1 iff. 0)1,0()0,1( and ).1,0(,0),( yxyx

(b) d satisfies P2 iff. yx ].1,0[, yx(b) d satisfies P2 iff. yx ].1,0[, yx

(c) d satisfies P3 iff. ),(),( txzy , if ,tzyx for all x, y, z, t [0, 1].

(d) d satisfies P4 iff. ),(),( xyyx ].1,0[, yx

Theorem 3.1 Let }...,,,{ 21 nxxxX and suppose that ]1,0[]1,0[]1,0[: XXd . Then

d satisfies P1-P5, iff. d is given by (2) for some map ]1,0[]1,0[]1,0[:

satisfying conditions (a)-(d) if lemma 3.2.

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Here, it is evident that the desirable function ]1,0[]1,0[]1,0[: in lemma

3.1 and lemma 3.2 is a restricted equivalence function. Thus, the knowledge of a restricted equivalence function gives an entropy measure of intutionistic fuzzy sets.

In earlier studies, entropy measures of intutionstic fuzzy sets have been defined using certain set of axioms. A well accepted set of axioms to define an entropy measure of intutionistic fuzzy set given by [Szmidt and Kacprzyk, 2001].and Kacprzyk, 2001].

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[Szmidt and Kacprzyk, 2001] suggested an axiomatic definition of entropy of IFE :

Definition 3.1[Szmidt and Kacprzyk, 2001] An entropy on ),(XIF the set of

all intutionistic fuzzy sets on X is a real valued functional ],1,0[)(: XIFE

satisfying the following axiomatic requirements:IFE1: 0)( AE iff A is a crisp set, i.e. 0)( iA xg or 1)( iA xg for all .Xxi

IFE2: 0)( AE iff )()( iAiA xhxg for all .Xxi

IFE3: )()( BEAE if BA i.e. )()( iBiA xgxg and ),()( iBiA xhxh for )()( iBiB xhxg IFE3: )()( BEAE if BA i.e. )()( iBiA xgxg and ),()( iBiA xhxh for )()( iBiB xhxg

or )()( iBiA xgxg and ),()( iBiA xhxh for )()( iBiB xhxg for any .Xxi

IFE4: ).()( cAEAE

In the next section, some examples of restricted equivalence and hence new measures of intutionistic fuzzy entropy have been presented.

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4.Some examples of REF’s and new measures of intutionistic fuzzy entropy

Some examples of REF are given as follows:

1. )1(loglog),(1 yxyx

yy

yx

xxyx

2. )1(log

)1()1(log

)1(),(

xyxyyxyxyx

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2.2

)1(log

2

)1(

2

)1(log

2

)1(),(2

xyxyyxyxyx

3.

1

4

1

4

1

)12(

1),(3

xySin

yxSinyx

4.

1

4

1

4

1

)12(

1),(4

xyCos

yxCosyx

5.

1

42

)12(

1),(5

yxCosyx

6.

12

11

2

1

)1(

1),( 2

1

2

11

6

yxyx

eyx

eyx

eyx

7. )1(2log)1(

1),( 11

7 yxyxyxyx

20

)1(

8.

)1(

1),(

||1

8

e

eyx

yx

Measures of intutionistic fuzzy entropy corresponding to REF’s ),(1 yx

to ),(7 yx had already been studied as listed in the following table:

REF Intutionistic fuzzy entropy Author (s)

),(1 yx

n

i iAiA

iAiA

iAiA

iAiA xhxg

xhxh

xhxg

xgxg

nAE

11 )()(

)(log)(

)()(

)(log)(

1)(

)( iA x

[Vlachos and Sergiadas 2007]

),(2 yx

n xhxgxhxg )(1)()(1)(1 [Zhang

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),(2 yx

n

i

iAiAiAiA xhxgxhxg

nAE

12 2

)(1)(log

2

)(1)(1)(

2

)(1)(log

2

)(1)( iAiAiAiA xgxhxgxh

[Zhang and Jiang, 2008]

),(3 yx

n

i

iAiA xhxgSin

nAE

13 2

)(1)(

12

11)(

2

)(1)( iAiA xgxhSin

[Ye,2010]

),(4 yx

n

i

iAiA xhxgCos

nAE

14 2

)(1)(

12

11)(

2

)(1)( iAiA xgxhCos

[Ye,2010]

),(5 yx

n

i

iAiA xhxgCos

nAE

15 1

2

)()(2

12

11)(

[Wei et al., 2012]

),(6 yx

2

)(1)(

16 2

)(1)(

)1(

11)(

iAiA xhxgn

i

iAiA exhxg

enAE

[Verma and Sharma,

22

2

)(1)(

2

)(1)(iAiA xgxh

iAiA exgxh

Sharma, 2013]

),(7 yx ))((2)()()()(log)1(

1)( 11

7 iAiAiAiAiA xxhxgxhxgn

AE

.1,0

[Verma and Sharma, 2014]

Now, we propose a new measure of intutionistic fuzzy entropy using REF ),(8 yx

n

i

xhxg iAiAeen

AE1

|)()(|1 1)1(

1)(

(4)

In order for (4), to be qualified as a suitable measure of intutionistic fuzzy entropy, it must satisfy the set IFE1-IFE4 of axiomatic requirements. In this paper, it has been proved that (4) satisfies IFE1-IFE4.

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paper, it has been proved that (4) satisfies IFE1-IFE4.

5. Concluding Remarks

In this communication, the study of intutionistic fuzzy information measure has been done in light of a newly defined special T-Norm and restricted equivalence function. It has been established that the knowledge of a REF give rise to a new intutionistic information measure. Further, a new intutionistic fuzzy information measure have been proposed. The validity of this measures has been tested axiomatically.

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(1986), 87-96.

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[3] De Luca, A. and Termini, S., A definition of non-probabilistic entropy

in the settings of fuzzy set theory, Information and Control, 20(1971),

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[4] Ebanks, B. R., On measures of fuzziness and their representations,

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