Fuzzy contactibility and fuzzy variables

Fuzzy Sets and Systems 8 (1982) 81-92 North-Holland Publishing Company

81

F U Z Z Y CONTACTABILITY A N D F U Z Z Y V A R I A B L E S Pei-Zhang W A N G Department of Mathematics, Beijing Normal University, China Received August 1980 Revised February 1981 This paper gives some advanced developments of possibility measure theory. There are two problems being solved in this paper: 1. What is the most essential property of the possibility measures? 2. What is the measure extension theorem concering the possibility measures? Finally, a structure which is analogous to the probability measure theory becomes more clear.

Keywords: Possibility measure, Probability measure theory.

O. Introduction Before I read the extremely important paper of Zadeh [1 ] I had considered such a problem: If we take a lot of elements from U forming a subset B of U, can we touch some elements in B which belong to a given subset A of U? The truth cf this fact can be represented by a mapping

O'A :~'(U)~{O, 1}, ~1 A O B ¢ 0 , O'A ( a ) --

(0.1)

|o Ana--¢

and we call era the contactability measure of A or contactibilitude of A. Now, if there is a fuzzy subset A instead of A occurring in this problem, how can we define the contactabilitude of A ? We see that (rA can be generalized to a mapping

[0, 1]: (0.2) u~_B

u~O

(in our papers, the fuzzy subset A was defined immediately by the mapping LI" U--->[0, 1], so that we used to write A ( u ) instead of ~ ( u ) ) . Therefore, we gave the following definition" 0165-0114/82/0000--0000/$02.75 © 1982 North-Holland

P.-Z Wong

82

Definition 0.1. If A ~ 4 ( U ) , the

contacmbilitude of

A is a mapping

or~ :19(U)--, [0, 1] which is defined by (0.2). I am happy to see that the definition of contactabilitudes is coinciding with the concept of possibility measures so that we can get another illustrative explanation for the possibility measures. The contactibilitude cr of a given subset has an important property:

cr(U B,)= V or(B,). IcT

(or.l)

I~T

Here, T is an arbitrary index set. When T is empty (or.l) becomes

or(0)= o.

(or.2)

We call a mapping or" ~ (U) ~ [0, 1] a fuzzy additivity measure if it satisfies (or. 1) (and (or.2)). We will prove a correspondence theorem in Section 5 which gives a one-toone correspondence between the additivity measures and the contactabilitude (i.e. the possibility measures). It implies that the most essential property of the possibility measures is the additivity (~.1). The source of trouble of the measurability problems in probability theory was the constraint on the extension of measures, but based on a fuzzy additivity measure extension theorem proved in Section 2 this constraint becomes very weak. The ample field defined in Section 1 will be in preference to the Borel field for the possibility measure theory. Fuzzy additivity measures are not the same as Sugeno-Terano fuzzy measures, but their important work was very helpful to me in this paper. The excellent work of Nahmias [3] had begun to constitute a structure which is analogous to the probability measure theory. In this paper we hope to make this structure more clearly.

1. Ample fields Set !9(U) = {A I A c_ u}, generally. Definition 1.1. An following properties:

ample tield M

(tn

I

on U is a subclass of ~ ( U ) having the

(A1) U e ~ ; (A2) If A ~ M, then A c e M; (A3) If A, ~ ~ (t e "l, T is an arbitrary index set), then

U A, ~ a~. t~T

Fuzzy contactability and fuzzy variables

83

These properties imply that 0 e ~ and (A4) A, e ~ (t e T)::> N,~T A, e M. Dellnition 1.2. Let s~ be an ample field on U; an atom containing u is defined by [u]&[uL A N {AIueA

ed}.

(1.1)

L e m m a 1.1

u

(1.2)

L e m m a 1.2. A is an atom in ~ if and only if O ~ A ~ ~ , and A is indivisible in M,

i.e. (VB) ( B e s ~ ::> B A A = O

or

BCAA=0).

(1.3)

Proof,, Suppose that A = [u], if (1.3) is not true, then there is an B ~ M such that

AAB#O#AABL

(1.4)

If u ~ B, then

u~AAB~,

(1.5)

so that [u]c_A OB. But A A B e t : 0 , so that [ u ] = A ~ A OB, this is a contradiction. If u ~ B c, there is a contradiction too. Inversely, suppose that 0 ~ A ~ M, and A is indivisible in ~t. Let u e M, then A n [ u ] # O , because of (1.3), so that A O[u]~=0, i.e. [u]=_A.

But, according to (1.1) we have [u]c_A, so that A =[u], A is an atom in ~ . The proof is finished.

!-!

L e m m a 1.3. For any u, v ¢ U, we have [ u ] O [ v ] = O or [ u ] = [ v ] .

Proof. Because [u] and [v] are atoms in M, by Lemma 1.2 they are indivisible. So the lemma is true. I'-I We denote u---v if [ u ] = [v], clearly, '---' is an equivalence relation on M. Thus U can be classified by ---, and we can write

Ua& UI--..

(1.6)

Theorem 1.1. The mapping

~,: ..~---,~ ( U a ) ,

A ~--~{a J3 u ~ A, c~ =[u]} is isomorphic.

(1.7)

84

P.-Z. Wang

Proof. ,p is an injection. Indeed, suppose that Ax ~ A2 (At, A 2 e ~ ) then there is a u ~ A l e A 2 . Without loss of generality we suppose that u ~ A 1 - A 2 . Because u ~ A~. we have [ u ] e ~0(A~), and because u¢ A2, it is easy to see that [ u ] ¢ ~0(A2). Indeed, ff [u]~ q~(A2), then there is a v ~ A2 such that [u] = [v], so that u ~[v] and therefore u ~ [ v ] D A ~ k O ; but t)~[tg]f'~A2~-O, this is in contradiction with Lemma 1.2. Consequently [ u ] e go(A0-,p(A2) :/: 0. For any B e ~ ( U ~ ) , set

A={ul[u]eB};

(1.8)

we have q~(A) = {[u] [ u e A } . From Lemma 1.3, we have t o ( A ) - B. Then ~0 is surjective. It is easy to prove that ¢ is isomorphic, l-7 Coroll~'y. A ~ st if and only if A is a union of the atoms in ~t: A-

U

[u].

(1.9)

uEA

For any subclass "-6'c O ( U ) , there exists an ample field ~t containing it: c¢ c ~t_c ~ ( U ) ,

(1.10)

because ~ ( U ) itself is an ample field. Clearly, we have: Lemma 1.4. If ,~, ( t ~ 7") are ample fields, then the intersection of them is an ample field. Definition 1.3. Let '6' be a subclass of .~(U), the minimal ample field containing c¢ (or, the generated ample field of c~) is

I

(ample field)}.

(1.11)

Property 1. (1.127 Property 2 . (1.13) Example 1. For R - ( - o o, +oo), we have [{{x} I x e R}] = ~ ( R ) .

(1.14)

Indeed, for any A ~ ~ ( R ) we have

a = U{{x} I x

A},

so that A e[{{x}J x ~ R } ] ; but [{{x}Jx~R}]c_~(R), so (1.14) is true.

(1.15)

Fuzzy contactability and fuzzy variables

85

Example 2. Let c¢ = { ( - 0 c x)[ x ~ R}, then [~¢] = ~)(R).

(1.16)

Indeed. for any x e R, we have

{x}=f'l

=

rt-----1

fqt-

,x)C.

rl=

By Definition 1.3 we have {x}e[W], so that {{x} I xE R}_c[g']. By Property 2 we have

[{x} Ix ~ R } ] c [[c6']] = [c¢], but [c6']_c~(U); from Example 1, we have [c¢]_c [{ix} I x ~ R}], so (1.16) is true.

Theorem 1.2. The class of all atoms in an ample lield s~ generates ~ : (1.17)

[U~,] = ~ .

Proof. Obviously we have Uac_s¢, but [M] = M, so that [Ua]_cs¢. For any A ~M,. We have A = [..J {[u]l u ~ A } , so that A ~ [ U a ] and therefore Mc_[U.~]. D

2. Fuzzy ad#2tivity measure Definition 2.L Let ~t be an ample field on U; a mapping o- : a/---> [0, 1] is called a [uzzy additivity measure on M, if it satisfies (o-.1) and (o-.2). In the introduction of this paper we said that the contactabilitude of a fuzzy subset o-a is a fuzzy additivity measure.

Definition 2.2. (U, ~t, or) is called a fuzzy field if or is a fuzzy additivity measure defined on an ample field ~ on U. Theorem 2.1 (extension theorem). Suppose that M is an ample field on U; given a mapping o-o: U~---> [0, 1]

(2.1)

there exists always an unique ]'uzzy additivity measure defined on s~ which satisfies o-([u]) = o-o([U])

(u e; U).

(2.2)

P.-Z. Wang

86

Proof. For any A ~ a/, set

or(A) -~ V oro([U]) t4~A

(or(0)=o).

(2.3)

We have

~([u])= v~[u] V ~o([V]). Because v¢[u], we have [ v ] = [ u ] by Lemma 1.3; we have obtained (2.2). Suppose that A, ~ a / ( t ~ T); by the corollary of Theorem 1.1 we have

A, = U {[u]l u~.A,}, so that

U A,= U t~T

t~T

{U{[u]lu~A,}}=U{[u]l

u~

UA,},

t~T

so that

~ (t~T U

At)

v

u A,}

t6T

t~T = V or(A,), tET

so that (or.l) is satisfied. Therefore the extension of oro is a fuzzy additivity measure. The proof of uniqueness is clear. U1

3. Induced fuzzy field Given a mapping

f.u--> v, an inverse mapping can be induced:

F'-~(vO--,~(u), f-'(B) = {u If(u)~ B} (B ~ ( V ) ) ,

(3.1)

and an inverse mapping can be induced from f-t"

r~ (f-')-'- ~(a~(o))--, ab(ab(V)), f(a/)={B

If-'(B)~}

(.~aXaXu))).

(3.2)

Obviously, f-1 conserves the set operations such as U, N, ¢, so that if a / i s an ample field then f(a/) is an ample field too. m

D e h i t i o n 3.1. Let (U, ,if, or) be a fuzzy field, f be a mapping f. U--> V;

Fuzzy contactability and fuzzy variables

87

we call (V, f(~/), 3) a fuzzy field induced by f, if 6(B)=or(/-~(B))

( B e f"(~)).

(3.3)

Clearly, 8 occurring in the definition is a fuzzy additivity measure. 6 is called a fuzzy additivity measure induced by or.

Lemma 3.1. Let c¢ c @( U) and f: U ~ V; we have =

(3.4)

The proof is not difficult.

4. Fuzzy variables Definition 4.1. Let ~ be an ample field on U, a mapping 6" U -* R is called a fuzzy variable on (U, a¢) if ¢~-'(B)a_.~

(B6.@(R)).

(4.1)

Theorem 4.1. ~: is a fuzzy variable on (U, ~) if and only if for any real x,/j-l({x}) is always the union of atoms in ~. Proof. Obviously.

I-1

Definition 4.2. Let (U, ~¢, or) be a fuzzy field, and 6 be a fuzzy variable on (U, ~); then the fuzzy additivity measure ore which is induced by or will be called the/uzzy distribution of ~. Set

p~(x)--A or~({x}) = or(l~-t({x})) (x ~ R);

(4.2~

p~(.) is called the fuzzy density of 6. Theorem 4.2. The fuzzy distribution of ~ is uniquely determined by the fuzzy density of !~. Proof. Obviously.

C]

5. Correspondence theorem Definition 5.1. 4 e ~F(U) is called a fuzzy subset on (U, M) if d : U"*[0, 1] is a fuzzy variable on (U, ,~). Set 3~(U, ~ ) ~ {d [ A is a fuzzy subset on (U, ~)},

(5.1)

~ ( U , ~¢)_a {or [ or is a fuzzy additivity measure on ~1}.

(5.2)

P.-Z. Wang

88

Theorera 5.1 (correspondence theorem). There is a one-to-one correspondence between ~(U, s~) and .a(U, af): 4 " .~(U, d ) ---,.a(U, a~)

(5.3)

such that every [uzzy additivity measure or ~ ~ ( U, a¢ ) is always a contactabilitude or" d = ~ - l ( o r ) ~ ( U , s ¢ ) , i.e. every [uzzy additivity measure is always a possibility measure.

Proof. Given A ~ :T(U, .d), we define (5.4)

,t,(A) = or~,

or~(B)& V d(u) ( e ~ ) , u~B

i.e. q~(A) is the contactabilitude of A. W e have

V ord(B,) = V ( V a A(u)) t~T

tET

u

t

t~T

or~ satisfies (or.l), so that ~ ( d ) = or,~ e ~ ( U, s~).

Inversely, given an or e ~l(U, s~), we define I/'(o')= d.,

(5.5)

(u ~ u ) .

d~(u)~or([u]) for any x ~ R,

dS'({x}) = {u Id(u)= x} = {u I , ~ ( [ u ] ) =

x}

= U {[u] I~([u]) = x}~ ~.

According to Theorem 4. I, A,, is a fuzzy subset on (U, ~). Note that

((~ o 4~)(d))(u) = (~,(4~(d)))(u) = (~(orn))(u) = ~([u]) =

V

ue[u]

d(v)= d(u)

(u ~ u),

Fuzzy contactability and fuzzy variables

89

so that

(q,' o 4,)(a) = d

(A e ,.~(U, ~)),

(5.6)

and

((4, o '#')(o-))(B)= (~(,P'(o-)))(n) =

V

(q'(cr))(B)

ttEB

= V o-([u])=o-(B)

(B e ~')

ue,[I

so that

( ~ o q')(cr)=cr

(o'e~t(U,~).

(5.7)

From (5.6) and (5.7), we have

and from this the proof of Theorem 5.1 can be finishzd.

F]

6. Product luzzy field Definition 6.1. Let M be an ample field on U and let ~ be an ample field on V: set

~ t x ~ a- [{A x B [A e ~ , B e~}].

(6.1)

We call ~ x ~ the product ample field of ~ and ~ . Suppose that D e ~ ( U x V), set

D,.A{vl(u,v)eD},

D.~a-{ul(u,v)eD},

(6.2)

and set

(,.~ x ~).. & {D,. I D e ..~ x ~},

(,.~ x ~).,, & {D.,, I D e,~ x ~}.

(6.3)

Obviously we have:

Lenuna 6.1. Let D, (t e T), D e ~ ( U x V); we have (6.4) lET

t

t

tET

t

teT

(6.5) D = C ¢ ¢:¢, (Vu)(D,,.=(C,.) ~) ¢:~ (Vv)(D., =(C.,~)c).

(6.6)

Lemma 6.2. Suppose that ~ , ~_~ ~ ( U x V); then •

¢:,

¢:,

(v,,)(c.,,

(6.7)

90

P.-Z. Wang

From (6.7), we have

[~],,. = [~,.],

[~].o = [~.,,].

(6.8)

Lemma 6.3. (,~ x ~),,. = ~ ,

( g x @).,., =,.~.

(6.9)

Proof. Let

~&{A xB I A e ~ , B e ~ } ; from (6.1) we have M x ~ = [~¢]. Clearly,

(.~ x ~),. = [~,],,. = [ % ] . Obviously, we have that rg.. = ~ , so that (,.~ x ~ ) , .

=[~]=~.

by analogy, we have (~x~8).,, = ,~. Theorem

6.1.

[(u,

v)]a×~=[u]~x[v]~.

l h r ~ t . Because [ u ] e ~ and [ v ] ~ ,

but

I-1

(6.10) we have

(u, v)e[u]×[v], and [(u, v)]= D {D I (u, v)~ D ~ ,~x~},= [ u ] x Iv].

If (6.10) is not true, then there is a u' e[u] such that ([u] + [v]),, _~[(u, v)]u,. ~: ~;

(6.11)

or, there is a v' e [v] such that ([u] x [v]).v, ~[(u, v)].v, ~=0.

(6.12)

Because of (6.11), we have

([u]x Iv]),,. =[v], and according to Lemma 6.3, we have

[(u, v)],,. ~. Therefore, [v] is not an atom in ~ which is a contradiction. By analogy, (6.12) gives a contradiction too. El Definition 6.2. Let (U, ~t, tr) and (V,~, 8) be two fuzzy fields and set (tr x 8)o([(u, v)])& tr([u])^ ~([v]).

(6.13)

Fuzzy contactability and fuzzy variables

91

According to the extension theorem, (6.13) uniquely determines a fuzzy additivity measure or x 6, which is called the product fuzzy field of additivity measure. (U x V, ~ x 30, or x 6) is called the product fuzzy field of (U, ~t, or) and (V, ~, 6).

7. Exle--~-"on prindple

Let (U, M, or) be a fuzzy field, and ~j, 71 be two fuzzy variables on (v, ~t). Denote

(IL rl)= T" U - . R 2.

(7.1)

Note that

~ ( R 2 ) = ~ ( R ) x ~ ( R ) = [ { A x B I A, B ~ ~(R)}]; because T - I ( A x B)=~-I(A)f'IrI-I(B)eM, we have ~ = { T - ' ( A x B) IA, B ~ ( R ) } c

M.

According to Lemma 3.1, we have T([~]) = [T(~)] = [{A x B I A, B ~I~(R)}] = ~(R2). But [~]c_.~, so that =_ Note that 'F(M)_c~(R2), so we have "F(.~) = ~(R2).

(7.2)

ortt~m):~(R2) --->[0, 1], ort¢,,1)(D)a-or(T-l(D)) (D e~(R2));

(7.3)

Define

then (R2,~(R2), or(¢.,o) is called the ~uzzy field induced by (~, rl). Definition 1.1. ~5 and rl are independent if the fuzzy field (RE,~(R2), or(~, rl)) is the product fuzzy field of the fuzzy fields (R, ~ ( R ) , ore) and (R, ~ ( R ) , orn). Theorem 7.1 (extension principle). Let 1~and rl be two independent fuzzy variables on (v, .~, or); then we have

V ~*y

where * = + , - , ' ,

(7.4)

= Z

+,...,and (7.5)

P.-Z. Wang

92

Proof.

p~..(z) = ff((f li¢ 'r] )- l({z}))

~r({u l (,f * n)(u) = z}) = ,,-({u I t(u).

n(,,,) = z})

=o-( U {ul~(u)=x,n(u)=y}) Xiy=Z $

l

XX i y = Z

=

V

X*y =Z

~({u I (6,

!

n)(u)=(x, y)})

= V o'
=

V

oq,.~,([x]x[y])

K*~=Z

= V (o-~([x]) ^ o-,,([y])) X * y =- Z

=

V

(p~(x)^p,,(y)).

El

X*~ =Z

References [1] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1 (1978) 3-28. [2] Wang Pei-zhang, Fuzzy sets theory and its applications, Shanghai, to appear. [3] S. Nahmias, Fuzzy variables, Fuzzy Sets and Systems, 1 (1978) 97-110. [4] M. Sugeno and T. Terano, Analytical representation of fuzzy systems, Fuzzy Automata and Decision Processes (North-Holland, Net York, 1975) 177-190. [5] C.V. Negoita and D.A. Ralescu, Representation theorems for fuzzy subsets, Kybernetika 4 (1975) 169-174. [6] A. Kaufmann, Inwoduction to the Theory of Fuzzy Subsets, %/ol. 1 (Academic Press, New York, 1975).