Fuzzy Sets and Systems 21 (1987) 147-157 147 North-Holland

M E A S U R E S O F F U Z Z I N E S S O F F U Z Z Y E V E N T S

Fermin SU/~REZ G A R C f A and Pedro G I L / ~ L V A R E Z Departamento de Matem6ticas, Universidad de Oviedo, Spain

Received June 1985 Revised December 1985

We prove some new results on two families of fuzzy integrals defined in a previous paper, and by means of them we obtain entropy measures of fuzzy sets (not necessarily finite) which contain as particular cases the measures of Batle and Trillas, and Weber. We define another measure of fuzziness which is a function of the previous ones and satisfies, with slight restrictions, De Luca and Termini's axioms.

Keywords: t-Seminorm, t-Semiconorm, Seminormed fuzzy integral, Semiconormcd fuzzy integral, Measure of fuzziness.

1. Introduction

Since De Luca and Termini [2], various authors have proposed measures of the degree of fuzziness or entropy of a finite fuzzy set [7, 14].

Knopfmacher [5] was the first to extend the measure of fuzziness to infinite fuzzy sets. Later Batle and Triilas [1], Kruse [6] and Weber [15] obtained other measures, using different types of integrals. In this sense, by means of two families of fuzzy integrals defined by Su~irez and Gil [11] which are a generalization of Sugeno's fuzzy integral [12] we present the definition of two families of measures of fuzziness which contain as particular cases the measures proposed by Batle and Trillas [1] and by Weber [15]. Unfortunately they do not satisfy all of De Luca and Termini's axioms [2]. In order to resolve this difficulty we define a third measure which is a function of the above two.

We also prove some news results on the families of fuzzy integrals defined in [11].

2. Preliminaries

Here v and A stand for maximum and minimum respectively.

Definition 2.1. Let (X, fl) be a measurable space. A function g:fl--~[0, 1] is called a fuzzy measure if:

g(~) = 0; g(X) = 1. (2.1)

I fA c B then g(A) <~g(B). (2.2)

0165-0114/87/$3.50 © 1987, Elsevier Science Publishers B.V, (North-Holland)

148 F. Sutlrez Garcia. P. Gil Altw~rez

If {A,}.,.~, is a m o n o t o n e sequence of sets in fl (A, cA~_~ or A , . . ,m A~ Vn • N) then

lim g(A~)=g{~rnA~}. (2.3)

Any classical positive measure g such that g(X) = 1 and the ).-measures of Sugeno [12] are examples of fuzz 3 , measures.

If fl is finite, m is called a possibilio" measure if r e ( X) = 1, m ( 0 ) = 0 and m(A U B ) = r e ( A ) v m(B) and b is called a consonant belief function [10] if b(X) = 1, b ( 0 ) = 0 and b(A N B ) = b ( A ) A b(B); these last two functions are other examples of fuzz), measures.

The triplet (X, fl, g) will be called a fuzzy measure space. A map f :X--- , [0 , 1] (or a fuzzy set) is called fl-measurable if the sets

H¢~ = {x I f (x ) t> a'} or H ~ = {x If (x) > a} are in fl for all tr in [0, 1]. We will denote by L(X) the class of fuzz), sets of X and by La(X) the class of

fuzzy sets of X whose membeship functions jr are #-measurable: f is then called a fuzzy event [18].

At some occasions we will. for better comprehension, represent the member- ship function of the fuzz), set ,4 by .uA. We will say that A is a crisp set if/z.a takes values in the set {0, 1} and/~a can be interpreted as the characteristic function of a classical set.

D e l i a i ~ n 2.2. If h is a fuzzy event and g is a fuzz), measure, the fuzz)" integral of Sugeno of h on A is defined by:

f , h g sup A g(A (2.4) O n Hi)]. a ~ O . 11

The main properties of this integral are given in [1.8. 12. 13]. There has been some criticism on the fuzzy integral of Sugeno, sometimes with false arguments, like in [16], where the author gives the following property (in order to assess that it is 'very insensitive').

Properly 2.3 [16]. Let v = ~-xh og. Then, for each function f such that Hr. = H~, we have:

~xf ° g =v.

That property is manifestly false as one can see in the next counterexample.

Connterexample 2.4. Let a e f0, 1] and let h(x )=a for all x e[O, 1]. Then ~-x h og=a . Let b e [ 0 , 1 ] with b>a. Then, if f ( x ) = b for all x e [ 0 , 1 ] ,

= X = H , h. Hence ~ x f ° g = a < b, which is absurd because the fuzzy integral over X of a constant is the same constant.

2.5. A triangular seminorm (t-seminorm) is a function T:[O, 1] ×

Measures of.fuzziness o f.htz.zy events

[0, 1]--, [0, 1] satisfying the following properties: (a) T(1, x) = T(x, 1) = x for all x of [0, 11. (h) If xt <~x,_ and Yl ~<Y2 then T(xt, y~) <~ T(xz, Yz).

149

The following functions are examples of t-seminorms:

r (x, y ) = x ^ y , (2.5)

x ^ y if x v y/> ),, for any Z in [0, 1]; (2.6) T._.~(x, y) = 0 i f x v y < ) . ,

T3(x, V) = ~ (xp AYq) if (X v y ) < 1 " L(X Ay) if (X v y ) = 1, p A q > 0 . (2.7)

Definition 2.6. A triangular semiconorm (t-semiconorm) is a function J. : [0, 1] x [0, 11---) [0, 1] satisfying:

(a) ±(x, O) = ±(0, x) =x for all x of [0, 1]. (b) Ifxl~<x2 and y~ <~ y2 then J-(xl, yl) <~ j.(x2, Y2).

It is easy to see that Z is a t-semiconorm if and only if there exist a t-seminorm T such that _L(x, y ) = 1 -- T(1 - x , 1 - y ) . The following funtions are examples of t-semiconorms:

±~(x, y) = x v y; (2.8)

J - , , ( x , y ) = { 1 v y f f X A y ~ < 2 ' for any ). in [0,1]; (2.9) . . . . i fx Ay >) . ,

~ ( x e v y q) i f x ^ y > 0 , - I -3(x 'Y)=[(xvy) i f x Ay = 0 , p A q > 0 . (2.10)

The t-norm and t-conorm concepts of Schweizer and Sklar, see [9], turn out to be special cases of the pre~4ous definitions.

3. Fuzzy integrats

Definition 3.1. Let h be a fl-measurable function (fuzzy event), T a t-seminorm and ± a t-semiconorm. We define the seminormed fuzzy integral of h over A ~B as:

fAh Tg sup [t~,g(A Hi) I , (3.1) T t') a~(O, 1]

and the semiconormed fuzzy integral of h over A as:

~ah g [ot, g(A n ~ ) ] . (3.2) l inf ± N . a ~ [ O . 1)

(3.1) coincides with Sugeno's integral when T = A and with the definition of Weber [15] when T is a t-norm.

The proofs of the following properties are given in [11].

150

Proposition

if

If

F. Sudrez Garcfa, P. Gil Alvarez

3.2. For all t-seminorms T:

hl <~h2 then fahlTg<~ fAh2Tg. (3.3)

A c B then ~AhTg<~fBhTg. (3.4)

fxa Tg=a for all a in [0,1]. (3.5)

fx(aVh) T g = a v f xhTg forallain[O, 1]. (3.6)

fAhTg=fx(hA[.LA) Zg. (3.7)

~xl~a Tg=g(A) for all A in [3. (3.8)

~ah Tz, lg<- fmh Tg<~ ~ah T~g. (3.9)

And for semiconormed fuzzy integrals we have:

Proposition 3.3. The fuzzy integral of Sugeno is a t-semiconormed fuzzy integral. For all t-semiconorms ±:

If hl <~h2 then f A h l Z g ~ A h 2 Z g . (3.10)

xa t g=a forallain [0,1]. (3.11)

f A ( a ^ h ) Z g = a ^ f A h l g forallain [0,1]. (3.12)

~,ah ±g= ~x(h ^ #a) ± g. (3.13)

xltA ± g = g(A) for all A in [3. (3.14)

If T and ± are continuous, we have some interesting results, like the following.

Theorem 3.4. Let T and Z be a continuous t-seminorm and t-semiconorm

Measures o f fuzziness o f fuzzy events 151

respectively and {h.},~N a monotone sequence h. 'r h, or h. J, h respectively. Then:

(a) ~ah, Tg ~ ~Ah Tg, (3.16)

or

~ah, Tg ~ ~a h Tg, (3.17)

respectively;

o r

fA t fA

ah, Z g ,~ fA h d_g, (3.19)

respectively; and in this way it is possible to extend the concept of fuzzy measure to fuzzy events by means of the above fuzzy integrals as:

f g(fi,) ~xlt,i T g (3.20)

o r I "

g(A) ]~x/~ _1_ g. (3.21)

T and Z must be continuous; otherwise the continuity condition of fuzzy measures (2.3) does not hold.

Counterexample 3.5. Let X = [0, 1] and g be the measure of Lebesgue in X. Let fi~ be a fuzzy event of X with membership function #,~ (x) = a for all x is X, and let {"t.}.~N be a sequence of fuzzy events with membership functions

a if x > 1/n, /z•.(x)= n . a . x ifx<<-l/n, w i t h a < l . (3.22)

Then, obviously, fi-n c fi~,+~ for all n in N and fi,, l'fi,. But if

~x I~a T2,~ g for any/~ of Lt~(X ), g(/~)

we have that g(.A) = a from (3.5) and

f g(A,) = t- /~.,i. Tz., g = sup ~. , [a, (1 - tr/n. a)] v sup T2., (tr, 0)

.Ix orE(0, a] tre(a, ll

= 0 v 0 = 0 for a l l n i n N .

Then lim,,._.= g(.,~,,) = 0 ~ a.

We will recall some definitions given by Batle and Trillas [1].

152 F. Sudrez Garcia, P. Gil Alvarez

If (X, /3, g) is a fuzzy measure space we say that N c P(X) is g-null if there exist F E /3 such that N c F and g(F) = 0. We say the space is complete if every g-null set belongs to /3.

From now on, we shall always deal with complete spaces. We say that h, and h2 are g-equivalent if {x 1 h,(x) # h2(x)} is g-null.

Proposition 3.6. Let g be a possibility measure. Then:

09 ~(h,vh,)Tg=fAh,TgvfAh,Tg;

(b) f, @I vWg=fAhl+,hdg.

Proof. (a) For all (Y in [0, 11, H2vh2 = H? U H?. Thus

(3.23)

(3.24)

c

! A (h, V hz) Tg = S;pI1 T[K g(A n (H? U H?))]

= sgppl] T [cu, g(A f-~ H:!) v g(A f-~ H?)]

= ~.y~~ T [a; g(A n &‘)I v s;P~~ T [a; g(A n @)I I

=fAWgvfAhJ’g~

Similarly for (b).

Proposition 3.7. If g is a consonant belief function, then:

(a) fA(hlAhZ)Tg=fAhlTgAfAh2Tg; (3.25)

(b) f, @I AWg=fAhlW+&- Proof. (b) For all (Y in [0, 11, HzlAhZ = HLhl fl Hkh2. Thus

(3.26)

f A (h, A h2) I g = ,$, I [cy, g(A n Hz’) fl g(A f-l Hz’)]

= i:ofl, I [CY, g(A n Hh”l)] A ,f:ofl, I [a; g(A n HkhZ)]

=fAh,lu+,lg. Similarly for (a).

Proposition 3.8. If hl and h2 are g-equivalent, and g is a possibility measure,

Measures of fuzziness of fuzzy events 153

then:

Proof.

(a) fahlTg=~ah2Tg;

(b) fAhiLg=fAh2lg .

Let M = {x I h~(x) 4:h2(x)}. Then g(M) = O.

fA hl Tg= fA [(hl AIIM) V(hlA~M~)]Tg

= fA(hl A~IM) Zg v fA(hl A~IM¢) Zg= ~AnMhl Zg v fAnMchl Zg

The same proof is valid for (b).

4. Measures of fuzziness

Several works dealing with measures of fuzziness [1, 5, 14, 15] use the function ('fuzziness function' [15], 'entropy N-function' [4, 6]) defined as: A:[0, 1]---> [0, 1], satisfying A(0) = 0, A(½) = 1, A(x) = A(1 -- x) and A strictly increasing in [0, ½].

From now on we shall suppose that A of e L~ (X) for every f e Lt3 (X). In [3] some properties are given of the partial order relation in L(X) defined

pointwise by

f<,g ~ ~f(x)<<-g(x) ifg(x)<<-½, tg(x) <-f(x) if g(x) > ½, for all x in X. (4.1)

Di Nola and Ventre [4] define

f<~2g ¢~' Ao f (x)<Aog(x)or , if Ao f(x)=Aog(x) , f(x)<~g(x) for all x in X. (4.2)

Also, Yager [17] states that the measure of fuzziness of a fuzzy set fi, is related to the distinction between the fuzzy set fi~ and its complement ,~c. This leads us to define the following relation:

Definition 4.1. L e t f ' = 1 - f f o r all f i n L(x). For all f and g in L(X),

f<-xg ¢* ]f(x)-f'(x)]>-lg(x)-g'(x)] f o r a n y x i n X . (4.3)

It is easy to see that the above definition is equivalent to

f~<l g ¢¢' If(x) - ½1 ~ Ig(x) - ½1 for any x in X, (4.4)

154 F. Sudrez Garcfa, P. Gil Alvarez

and (4.2) is equivalent to

f<~2g ¢:) If(x) - ½l > [g(x) - ½l or, if I f ( x ) - ½1 = Ig(x) - ½l,

f (x) <<- g(x) for any x in X.

The following proposition gives the principal properties of ~<1.

Proposition 4.2. For all f, g in L(X): (a) I f f<~l g then f ' <~1 g'. (b) Iff<~lg thenf Af '<~g Ag' a n d f v f '>~g vg ' . (c) I f f < ~ g t h e n f A½<~lgA½ andgv½<~l fv½. (d) Iff<~lg thenf Af '<~lg Ag' and f v f '<~lg vg ' . (e) I f f<~lg then f<- l f A g<~lg and f<~lf v g<~lg. (f) I f f<-' g then f<~lg. (g) I f f ~ 1 g and f, g are related by <~' then f <<-' g. (h) f ^ ½ < l g ^ ½ and g v ½ <~f v ½ if and only i f f < g . (i) Iff<~lgthen A o f<~A o g.

(4.5;

5. N-measures

Knopfmacher in [5] extends the measures of fuzziness to infinite fuzzy sets; Batle and Trillas [1], Kruse [6] and Weber [15] make use of different integrals to define measures of fuzziness. We agree with Batle and Trillas when they say in [1] that fuzzy integrals are the adequate tools for defining measures of fuzziness for infinite fuzzy sets. Accordingly, we give the next definition:

Definition 5.1. Let T be a t-seminorm. We define the N-measure of fuzziness of A e Lt~ (X) as

dr, A,g(fi,) = fx (A o iz,~) Tg. (5. 1)

We shall simply write dN(fi0 when no confusion is possible. If T = ^ in (5.1) then dN is a Batle-Trillas measure.

We state the main properties of this measure.

Proposition 5.2. (a) I f A is a crisp set then dN(A)= 0. I f dN(A)= 0 then g(b, = ½)) = 0.

(b) Iflz,~ <~11t~ then dN(A) ~< dN(/~). (c) / f #,~ = ½ then dN(fi~) ~> aN(B) for all B of Lt3(X ). ( d ) =

(e) I f dN(,~) = 1 then g({/La = ½}) = 1. (If g is a consonant belief function then Iz,~ is g-equivalent to ½.)

(f) I f g is a possibility measure then g-equivalent fuzzy sets have the same measure of fuzziness.

Measures of fuzziness of fuzzy events 155

The proof is straightforward. However, the inverse of the first implication of (a) (if dN(A) = 0 then A is g-equivalent to a crisp set) is not valid:

Counterexample 5.3. Let X = {0, 0.1, 0.5, 1} and let A be a fuzzy set of X with membership funtion p a ( x ) = x, and A(x)= 2. x for all x in [0, ½]. Let g:P(X)---> [0, 1] such that g({0}) = g({0.5}) = 0, g({0.1}) = 0.5, g({1}) = 1 and g(B U C) = g(B) v g(C) for all B, C in P(X). Then

I X if ~ = 0 ,

H~ °"~= {0.1;0.5} i f 0 < o c ~ 0 . 2 ,

[. {0.5} if c~ ~> 0.2.

Thus

g(Ha~ °~'~) = .5 if re=O, if O< tr ~< 0.2,

if 0¢ ~> 0.2.

Let T = T2,1 be defined as in (2.6). Then

dr2.,(-A) = sup T2., [o~, g(H~*~'a)] = 0 v T2,1[1, g(Ha*~'a)] = Tz,,(1, O) = 0 ae(O, 11

and g({x I {0, 1}}) = g({O. 1; 0.5}) = 0 .5 ,0 ."

Thus fi~ is not g-equivalent to a crisp set.

6. C-measures

In the previous section we have defined a family of measures that do not satisfy all of the axioms that De Luca and Termini impose on a measure in order to be a measure of fuzziness.

However, in this section we will define another family of measures of fuzziness by using semiconormed fuzzy integrals, since we still believe that fuzzy integrals are the appropriate tools for defining measures of fuzziness.

Definition 6.1. Let _L be a t-semiconorm. We define the C-measure of fuzziness of A by

~x(A o ~t;t) l g. (6.1) d±;,,,g(A)

We shall simply write dc(A) when no confusion is possible. If 3_ = v then dc is a Batle-Trillas measure [1]. The main properties of this

measure are given in the following proposition.

Proposition 6.2. (a) I f A is a crisp set then dc(A) = 0. Conversely if dc(A) = 0

156 17. Sudrez Garcla, P. Gil Alvarez

then #A is g-equivalent to a crisp set. (b) I f I~A <<-1P~ then dc(,4) ~< dc(/~). (c) I f IzA -- ½ then dc(A) ~> dc(B) for all B o f Ltj(X). (d) dc(A) = dc(A¢). (e) I f dc(A) = 1 then g({x IliA(x) ~ {0, 1}}) = 1. (f) I f g is a possibility measure then g-equivalent f u z zy sets have the same

measure o f fuzziness.

It would be convenient if these measures satisfied: if d c ( A ) = 1 then /~A is g-equivalent to ½. Unfortunately they do not:

Cotmterexample 6.3. Let g(A) = 1 if A ¢ 0, and g(0) = 0, and let X = {0, ½, 1}. L e t / i be in L~(X) with I~A(x) = x and let A(x) = 2. x for all x in [0, ½]. Then

n "A°u~ = {x [ zi o g~i(x) > ~} :/:0 for all a ' e [0, 1).

Thus g ( H "a°"a) = 1. This implies that

dc(fi) = -~ (A o g,~) _1_ g = inf _1_ (o~, 1) = _1_ Jx ,re[O, D

for any t-semiconorm 1 , and

g({x I p~(x) ¢ ½} = g({O, 1}) = 1 ¢ O.

7. O-measures

In order to obtain measures that satisfy De Luca and Termini's axioms [2], we define the following.

Definition 7.1. Let 0 be a function 0 :[0, 1] x [0, 1]--> [0, 1] such that

O ( x , y ) = O ¢:> x = y = 0 , O ( x , y ) = l ¢:> x = y = l ,

increasing with respect to both variables. We define a O-measure o f fuzziness o f i t by

D e ( d ) = O[dN(A), dc(A)]. (7.1)

The main properties of this measure are gathered in the following proposition.

Proposition 7.2. (a) I f A is a crisp set then Do(A) = O. Conversely if Do(A) = 0 then lz;~ is g-equivalent to a crisp set.

(b) I f PA <<-xl~h then Do(A) <~ Do(B). (c) I f I~a =- ½ then Do(A) = 1. Conversely if Do(A) = 1 and g is a consonant

belief function then I~z is g-equivalent to ½. (d) Do(d) = Do(At). (e) I f g is a possibility measure then g-equivalent f u z zy sets have the same

measure o f fuzziness.

Measures of fuzziness of.fuzzy events 157

8. Conclusion

In this paper we have used the function of negation n ( x ) = 1 - x . However, it is also possible to use a general function of negation n ( x ) as in [15].

The aim of this paper was to define a family of measures of fuzziness which generalizes the Batle-Tril las measure [1]. The advantage of having numerous possible measures of fuzziness allows a user to choose the one that is most convenient for his particular requirements.

References

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[14] E. Trillas and T. Riera, Entropies in finite fuzzy sets, Inform. Sci. 15 (1978) 159-168. [15] S. Weber, Measures of fuzzy sets and measures of fuzziness, Fuzzy Sets and Systems 13 (3)

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