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Set theory for fuzzy sets of higher level
Fuzzy Sets and Systems 2 (1979) 125-151 © North-Holland Publishing Company
SET THEORY FOR F U Z Z Y SETS OF HIGHER LEVEL Siegfried G O T T W A L D Sektion Mathematik, Kari-Marx-Unit, ersitiit, 701 Leipzig, German Democratic Republic Received January 1978 Revised May 1978 We extend the usual notion of fuzzy set in such a way that the elements of fuzzy sets again can be fuzzy sets. For such fuzzy sets of higher level the fuzzy set theoretic operations are generalized up to the notion of a fuzzy mapping. In our presentation of the results we use a suitable many valued logic, indicating in this way the close formal connections between fuzzy and classical set theory.
1. Introduction Fuzzy sets are the mathematical models for extensions of vague notions. They are characterized by membership functions which permit grades of membership between full membership and non-membership. It is appropriate to consider these membership grades as truth values of a suitable many valued logic, and to develop the theory of fuzzy sets in the language of such a logic (cf. e.g. [2, 3]). We confine ourselves to the real numbers between zero and one, as was done by Zadeh [15] a n d since then in most papers on fuzzy sets. Generalizations are possible, but we will not consider them here. We use in this paper logics which often are called fuzzy logics. They are versions of the many valued logics of Lukasiewicz [12]. Some remarks on these logics are given in [8]. Their truth value sets are W, = [0, 1] and Wm= { k/m- 1]0 < k < m - 1 } for every natural number m > 21" their connectives -7 A 1, A 2, V t, V 2, -" ~-* are defined by" - l s = l - s , SAlt=min{s,t}, SA2t=max{O,s+t--1], s v t t - max {s, t}, s v 2t = min { l,s + t:,, s---,t = min { 1, 1 --s + t}, s~t = 1 --Is - t[. Of course, we have defined only their corresponding truth value functions, using always the same denotations for connective and function. Later on we use also ^ , or v ,, if in f;ome expression H both A 1, A 2 or both v 1, v 2 are permitted. If H laa,'~, more than one occurrence of the index ~, then it always has to be interpreted in the ~ame way. [ H I is the truth value of H. Bounded quantifiers V~, 3M, M a nonempty class of classical set theory, are defined by [VMxn(x) = inf {[n(a)][ae
[3 MxH(x)] = sup
M},
{[H (tl)]la e M ].
These logics we denote by Lm, m > 2, and L , " if only some unspecified of them is considered we use L resp. Wwithout index. By W ÷ we mean W~{0}. ~For m = 2 all is reduced to classical logic and classical set theory. 125
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Usually fuzzy sets are defined over some "universe of discourse"--which itself is a classical set. But not always such an universe of discourse can be a set of "wellknown" things. If one considers e.g. vague notio~is of higher order--i.e, properties of notions, relations between properties of notions etc., e.g. the vague notions "is a vague notion" itself--the corresponding fuzzy sets must be such that their "'elements" themselves are fuzzy sets. Therefore fuzzy sets of higher level are interesting objects to ~tudy. One can expect applications e.g. in linguistics (cf. [14, p. 40]). The technical problem posed by that discussion is the construction of an universe of discourse which is closed v.nder the formation of fuzzy subsets. We attack this problem by construction of a cumulative system of fuzzy sets. Such systems have already been described b3 Klaua [10, l l ] and the authoc [5]. But Klaua's system has the disadvantage that one is not able to develop it fully within the above mentioned many valued logic because of lack of a suitable fuzzy identity. The basic ideas for the present construction ,tre given i.n [7-1.2 Starting with a set R(0) of urelements, one defines simultaneously by transfinite recursion rank sets R(=) of fuzzy sets of level at most =, the fuzzy membership predicate e, a fuzzy rank relation i - , and fuzzy identity - . E = ~ =~0,fl(~) is the class of all fuzzy elements, M=E'~R(O) is the class of fiizzy sets. We use a,b,c,x,y,z,.., as variables for fuzzy elements and A,B,C,X, Y,Z,... as variables for fuzzy sets. Unbounded fuzzy quantification means bounded fuzzy quantification over E. The fuzzy rank relation has truth value one, [x i--y] = l, iff there does not exist an ordinal fl with yeR(flD and xCR(fl), i.e. iff the level of x is not greater than the level of y. Each R(~+ 1) is the union of R(a) and the set of all those functions .f : R(~)~[0, 1], for which for every fl<~. there exist x,7 with f l < 7 < ~ , x~R(7), and f (x) ~ 0, and for which for all x, y e R (~)
[x-y] A 2f(x)Sf(y). R(2), 2 limit ordinal, is the union of all R(fl), fl<2. Fuzzy membership is defined by (*)
[xeY]={~ (x) if x~d°(y)'otherwise"
with do(y) ~he domain of function y, doly) empty for y urelement. And fuzzy identity is in the case of x~R(O) or y~R(O): i_x=y]=f~
if x--),, otherwise,
and in the case of x, y ~ M:
[x-y]=[Vz(ze, x~-~z~y) A
2 x [ - y A 2Y[-- X].
2Both [7] and the present paper are part of the authors Habilitationsschrift 16].
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As an auxiliary notion let us introduce a f u z z y equality .~ for fuzzy sets by [A ~ B]=[Vx{xeA~--~xc, B)]. Then we have always 3 a -bA2be.A~aeA, a -b.--~Vx(aex~-~be..x), A .~ B,-.Vx(xe A~-~xr, B ), A - B , - - - , A , ~ B A 2 ( A ~ B A 2B[-- A). Our construction of the cumulative hierarchy of fuzzy sets has some analogies with the construction of Boolean valued models for (classical) set theory (cf. e.g. I-9, p. 53]). But there, the Boolean membership values have a much more complicated definition as our (,), the same is the case for Boolean identity" it is an open question if that way out will be helpful for the theory of fuzzy sets. We prefer (,) as we wish to identify the membership grades with truth values of fuzzy membership. Using the vocabulary of our logics L and r., =, E , ~ as binary relation constants, we can build up well-formed formulas in the usual way. If H(x) is such a wff with free variable x and with properties (i) for all fuzzy elements a, b there holds a-b
^ 2H{al-.H(b),
(ii) there exists an ordinal ~ such that for every fuzzy set bC~R(:~) the expression HIb) has truth value zero, then there exists exactly one fuzzy set A with property Vx(xc, A.-.H(x)).
{xllHtx)l
Let us writ,J for this fuzzy set A. As special fuzzy sets we consider the empty.litzzy set 0, characterized by [ x e 0 ] = 0 for every x e g ; fuzz)' singletons {al, for each truth value t and each fuzzy element a, characterized by [ x e { a ' j , ] = [ x - a ] ^ 2 t : and jit_-zy rank sets .~1~, characterized by Ix e ~/~] = 1 iff x ~ R{~) and [x e, .~#~] = 0 otherwise. We write {al for {a} 1" In [7] we define also ajitzzy inclusion relation ~_ by A ~_B.-~Vx(xe, A ~ x e , B ). heor ~ as a corollary we have A ~ B*-,A c_B ^ I Bc_A. 3If we state a fuzzy expression H in this way, this is short for the statement [H] = 1.
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With every fuzzy set A we correlate a classical set, its support
IAI = {x ~ E I I:xe A] 4:o}. By =~, ~ , A , V we denote implication, biimplication, generalization, and existential quantification of classical logic.
2. Elementary algebra of fuzzy sets Definition 2.1. For fuzzy sets A, B let
Au,B=dr{XllxaA v ~x~B}, An,B--df{Xllx~A A tx~B}. These definitions are correct ones, since for every y: y ~ x A 2 ( x g A *1 x e B )
~ ( y = x n 2Xe A ) * t ( y - x n 2Xe. B) ~ y e A "1 Ye B with , for ,\, v. But it is not l:..ssible to define in this way e.g a second conjunction with A 2 instead of A ~, because not always [y=--x A 2 { x c A A 2 x s B ) ] < [ y e A
A 2ysB].
Later on we will find another way to define corresponding operations for fuzzy sets by ^ 2, v 2. As an easy consequence of Definition 2.1 we note
AGB~AutB~B~AntB~A.
(*)
For the next results, it is convenient to change the formal definition of a lattice slightly, using in all defining equations (distributivity included) ~ instead of -=. Because for fuzzy sets A,B, A = B iff A ~ B--i.e. iff [A ~, B ] = 1--for systems of fuzzy sets the modified definition is equivalent to the usual one. Now we have
Proposition 2.2. For every ordinal ~ a distributive lattice with smallest and largest element is given by ['R(~ + 1 )~-R (0), n l , u t, O, ~d~].
Furthermore [M, "~t, wl,O]
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and also JR(2 )"~R (01, n , , u 1,0], for limit ordinals 2, are distributive lattices with smallest element.
Proof. By easy computations. From (,) now we see that the relation c_ is the right lattice ordering for these lattices. This gives us some monotony properties for n t, w l with respect to ~_, but we can prove more generally" A ~_B~Ac~I C~_Bc~I C, A ~_B---,A u t C ~_Bul C. As a corollary we have the full many valued substitutivity for fuzzy equality: A ~ B - - , A n t C ~ B n I C, A~.B~AuIC~Bu1C. ( o r r e s p o n d i n g results for - instead of ~ are true because for - there hold many valued versions of the well-known identity axioms of logic (cf[7]). One proves also A~_B A tA~_C~*A~_Bc~ IC, A~_C A IB~_C~-~AutB~_C. It is easy to extend these results to corresponding ones for a second fuzzy inclusion, defined by A~_*B*~,A~_B A 2 A F B . Furthermore, it is possible to prove yet more set algebraic fuzzy properties by using results of fuzzy propositional logic. We mention only the fuzzy implications A c_BoA ut
(BnIC)~.(AuIB)c~IC,
(AntB~Ac~tC) 2 A2Au1B~AutC~B~C" in the last expression with (H) 2 short for (H A2H), H the expression "Ac~tB ~An 1 Later on we need also xF" AutB*-'*xF'- A v l x V B, xF- AnlB.-*xV" A A lXF" B.
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Now, the notions of generalized union and generalized intersection shall be defined--however, we will do it not for fuzzy sets of fuzzy sets, but for systems, i.e. classical sets, of fuzzy sets. Definition 2.3. Let M, N be systems of fuzzy sets, N non-empty:
In both cases, ix IIH(x )} is a correct description of a fuzzy set; e.g. always there holds (Q for 3, vl
= - x A 2Qny(xs y ) ~ Q u y ( z - x
A 2XSy}
--,Qny(ze y). Clearly. always
Atu,B=U{A,B },
Am,B=~{A,B}.
For systems M, N of fuzzy sets and fuzzy sets A one can also prove e.g.
U(M,-,Nt UM,-,,UN, VuX(XcA}~UMc_A, M subsystem
of N =~U M c_ L_)N ;
for nonempty systems M, N one has also
VMX(A~_X)--)A~_(~M, M subsystem of N=#,(~N ~_N M . More details are proved in [6]. As remarked in [7], for certain A the fuzzy predicate "x~_ A" does not define a fuzzy set. Hence there we defined the.~zzy power set by* PA=,jf{x[ix ~*A}.
Proposition 2.4. (i) A ~_* B,-,PA c_PB, (ii)
PALJtPBGP(A~IB),
(iiit
P(Ac~IB)GPAnlPB.
4For urelements x we put [x_ *A] =0.
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Proof. (i) By transitivity of __, for every x A~_B~(x~_A~x~_B).
Assuming A E B, it is easy to get (x~_A---,x~_B)~(x ~_* A--.x ~_* B).
Both fuzzy implications give A ~_*B--*PA~_PB.
The reverse fuzzy implication is obvious. For 0i) one can use the fuzzy implication x~_A v l x G B ~ x ~ _ A u ~ B ,
and for (iii) t h corresponding one, x~_An~ B--,x~_A ^ ~x~_B.
Corollary 2.5. (i) A E (i'.')
B,-,PA E PB,
PA ~ PB,-,PA c_ * PB.
Proof. (i) By definition of the fuzzy rank relation we have 5 PAEPB*~
A
V (XEY),
X~IPA I YEIPBI
and also PA E P B ~ A E B because of A~IPA I and B ~ I P B I. Furthermore AE B ~
/~ (X E B)=:,PAE PB. X~IPAi
Hence A E B'~PA E PB,
and this classical biimplication is equivalent to the fuzzy one, for :X E Y can have only truth values zero and one. (ii) one proves using (i) and Proposition 2.4 (i). SRemember footnote (3), and hence that "PA E PB¢.,.'' "" is short for "IPA E PB] = 1 ~ " "'.
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In classical set theory, Proposition 2.4 (iii) is an equation. In the fuzzy case one cannot prove the second inclusion, needed for equality, because only [ P A n 1 P B ~_P{A n l B)] 4=0
(,)
is true. Clearly, if [X e P A n ~ P B ] = 1 for some X, X ~_*A and X ~*B, hence X * A n~ B. Therefore {,). But for any tlath value r > 0 one can choose three fuzzy elements a, b, c of equal rank with
[ a - b ] = [b-=c] = [a--c] =0" put for Y={a}" A~=Yw,{Y},., B~=Yu,{{b}}~, and X = Y u , { { c } } , . Now A , n ~ B r = Y and hence IX ~_*A,]=[X ~ * B , ] = l - r . On the other hand [X E A~n~ B , ] = 0 and therefore IX ~_* A,c,, B,] =0. All together this gives [PAr n l P B , ~ P(A,n I B~)] < r. Hence (,) is sharp. 3. Generators of fuzzy sets
It is an essential property of our fuzzy sets of higher level that there exist fuzzy singletons A={al=:xllx-al with card Ihl>l. Hence need not know all elements of the support IAI or a fuzzy set A for a description of A, but only a suitable subset of them. Definition 3.1. A generator of a fuzzy set A is an ordered pair (E, r) of a subset E of IAI and a function r" E ~ W + such that A=
~
E}.
A set E is a generating set of A iff there exists a function z such that (E, z) is a generator of A. Proposition 3.2. (i)(E,x) :s a generator of A iff z "E--, W ~ and
/~ i [ a e A ] = s u p ( [ a - x ] a
.~E
^ 2fix))).
E
(ii) A set E is a generaling set of A iff there hold
Va(as A*--~:iEx(a==-x ^ 2xaA)).
(,)
Proof. (i) is obvious. (if) If E is a generating set of A, consider any generator (E, z) of A. For in this case always z ( x ) < [ x e A ] for x e E and also [ a = x ^ 2 x e A ] <[aeA], (,) is an easy consequence of (i). Now suppose (,). Define z ' E ~ W + by z(x)=[xeA] and apply (i) to (E,z).
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Corollary 3.3. Assume that E is a set with property
Ax
^ 2yra]).
yeE
Then E is a generating set of A and A 4=0.
Proof. By Proposition 3.2 (ii).
Proposition 3.4.
Suppose that EI,E 2 are generating sets of the filzzy sets A I,A 2.
Then there hold A 1 [-'- A2~""~VEIX 3E2Y(X[-" y).
Proof. Straightforward from Definition 3.1 and the construction of the fuzzy sets.
Proposition 3.5. (i) (ii)
Let E l , E 2 be generating sets for A I , A 2 . There hold
A 1 ~A2*-...}VEtX(XI5 A l ~ x 8 A2) , A l .~ A2~"-*VEI,.,E2X(X8 A,,~-.,xe Az).
Proof. (ii) is an easy corollary of (i). The fuzzy implication Al GA2-}VF.,x{xsAI'-*xsA2)
is obvious. The fact that there hold x e A 2 A 2y=--X-"*ysA2
for any x, y and therefore xsA2-..~ysA 2 v 2y~x
holds too, proves (xgAI~xgA2).-o{xgA,-.*ysA2
v
2y~x).
Hence there hold VEIX(xeAI"'*xsA2)~(3F.~X()'=-x
^ 2Xg A1)~yr; A2).
Now we apply Proposition 3.2 (it) and fuzzy generalization with respect to 3' and get VE,X(X8 A 1 -"*x15 A2)--*A 1c_ A2"
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Corollary 3.6. For fuzzy elements a, b,c,d and truth values s, t there hold 6
{a}~___ {b}, ,-~(a=-b ^ 2t) v 2 ~, Oil
{a,b}.,~_ {c,d}, , - . ( ( ( a - c v , a ~ d ) ^ ~( b - c v ~b - d ) ) ^ 2 t ) v
2 ~.
Proof. By straightforward computations. We apply this result to some more special cases. The case s = t yields e.g. the following simplification: {a}~_ {o}~ ,--~a=b v ,~--~{a}~.v, {bJ~. Later on we are also interested in: {a,b}c{c,d}~--~(a=-c v 1 a - d ) A , ( b = c v ,b=-d),
{a,b} ..~ {c,d}.'.-,(a-c A~ b = d ) v ~(a=d ^ ,b=c). Finally we mention the following coincidences of fuzzy identity and fuzzy equality:
{al {a,b} ~{c,d},-~{a,b}=_{c,d}.
Definition 3.7. Suppose (E,r) as a generator of A. We call (E,r) a canonical generator of A iff A x ~ E ( r ( x ) = [ x c A ] ) ; and,we call (E,~)a minimal generator or a basis of A iff E is a minimal generating set of A with ~espect to classical inclusion. Proposition 3.8. Every basis of a fuzzy set A is also a canonical generator of A. Proof. Let (E, 3) be a basis of A, and assume that for some Xo ~ E there hold Z(Xo) < [ x o e A ] . Consider E ' - E ~ { x o } and z ' = z [ ' E ' . Because there hold [xoeA] =supy~E max {0, [Xo = y] + z ( y ) - 1} it exists some Yo e E with property r(Xo) < max {0, [Xo - Yo] + z ( y o ) - 1} and hence with {Xo},~xo~{Yo},tyn~. Therefore also (E',~') is a generator of A, contradicting the minimality of E. Hence (E, z) is canonical. For every fuzzy set A the support IA[ is a generating set of A; if one defines z'[A[-~W + by z ( x ) = [ x e A ] , then (IAl, ),s a canonical generator of A. Hence every fuzzy set has a canonical generator. But does there exist always a basis? The answer is affirmative for the finite-valued cases and regative for the infinitevalued one. 6To have short formulations, we use "s", "t" also as constants of our language, g is short for "-Is.
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T h e o r e m 3.9. [L~]. There exists a fi~zzy set without any basis.
Proof. We choose a sequence (r,),,~n of truth values, 7 which is monotone decreasing and converges to zero. We choose also some b ~ E with OF- b. Let be A = ~{{{b},.} 1 _ , . I n , N } . Suppose, E is a basic set of A. E must be nonempty and every m ~ N. Because for every n < m ~ N,
:#{{b j~rmJ"~ for
E{b}, - { t , } . . ] = we have for every k ~ N
Take atE. Assume [ a e A ] < l . Consider E'=E'~{a}, choose k ~ N > [ a ¢ A ] ; obviously, for every n > k and B=(,.J{{c}t~,ajlc~E', ---
with 1 - - r k
~°
[{b},..~; B] = 1 - r,, = [{b}, ~;A], hence A=B, and E is not minimal. Thus [ a e A ] = 1 for every aeE. Also, if a,a' eE, ~ ~ have limit 1 Hence a = a ' the sequences ( [ a = { b j,,]),,~ and ([a'={bj~,,]),,~ and E is a singleton: E = {aj. For this a, lim,_. [ a = { b j~j = l , hence 01-a. But already lirn,~, [x~. {bi~,] = 1 has x = 0 as only solution. Thus no such a exists, A has no basis. Proposition 3.10. [L,, n finite]. For each nonempty family (E:),~! oJ" generating sets
of a fuzzy set A also their intersection ~ I E ~ is a generating set of A Proof. Put E*=(']i~tEi. If A = 0 then always Ei is empty, and hence also E*. In this case E* is a generating set of A. Furthermore suppose A 4:0. Then our proof is finished if for E* there hold the supposition.,_, of Corollary 3.3. Therefore choose any fuzzy element a. 'We have to show the existence of some v e E* with
[a~A]<[a--y n 2ye, A] ~. Case 1" a ~E*. There is nothing to prove" take y = a . Case 2: a~Ui~lEi~E *. There exists j ~ l with a~Ej, and k1~I with aCEk,. Furthermore, there exists x~ ~ Ek, with |
[as A] = m a x {O,[a=xl]+rk,(Xl)-- 1,,
°
first, according to Proposition 3.2. (i) the value [a8 A] is given as a supremum, but this supremum here has to be a maximum. Hence we have
[ae, A]<[a=-xl n 2xtaA]<[xIIzA]. 7N is the set of natural
numbers.
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We are done if x t ~ E*. Otherwise there exist
k2 E I
with x~ ¢ Ek2, and x2~Ek2 with
[Xlt~A]'<[Xt--x 2 A 2 x 2 8 A ] < [ x 2 8 A ] and with
[ a s A ] = [ a = x l A 2Xt =--X2 ^ 2 x 2 8 A ] < [ x = x 2
A 2x28A] •
The last equality holds true, for we have
[a=--xl
A 2Xl ~ x 2 A 2 x 2 ~ , A ] < [ a = x l
A
2xt~,A]<[a~,A]
by the properties of = and e, and by the choice of xt, x2 also
[ae, A ] < [ a = x l A 2 x t r . A ] < [ a = x l /', 2xl =x2 ^
2x28A].
Anew we are done if x2 ~ E*. Otherwise this procedure continues, and produces a sequence x~, x2, x3 .... with
Ix, ~ A] <[x2e A] < [x3e A] < " " which therefore must be finite, if the procedure stops with Xk, then Xk ~ E* and for y-~ Xk the desired property holds. Case 3: a~[_)i,~Ei. There exist j e l and z~_Ej with [aeA]<[a=--z A 2 z e A ] . By case 2 there exists some y e E * with [ z e A ] < [ z = y A 2ysA]. Hence this y has also in this case the desired property.
Theorem 3.11 [L,, n finite]. Every fuzzy set has exactly one basis. Proof. Consider the system 9Jl of all generating sets of a given fuzz) set A. 9J/is nonempty and (']gJie~JJl by Proposition 3.10. Hence (']gJl is the only minimal generating set of A. In addition to the existence of bases another interesting question concerns the existence of canonical generators. We have seen that each fuzzy set has a canonical generator But does each generator (E, z) of a fuzzy set A contain a canonical generator--in the sense that for some subset E' of E the pair (E',r I'E') is a canonical generator of A? Again the answer is affirmative for t h e finite-valued cases and negative for the infinite-valued one. For L ~ we cot~sider the fuzzy set A which was constructed in the proof of Theorem 3.9. Put t r ( x , ) = l - r n for every l l < n ~ N . It is easy to see that ({x,] 1 < n ~ N l , a ) i s a generator of A, not containing any canonical generator.
Propositiq,n 3.12 [L,,, n finite]. For every generator (E,r) of a fuzzy set A there exists a stibset E' of E such that (E', z ~E') is a canonical generator of A. Proof. Let a generator (E, z) of A be given, and consider the subset
E ' = L ~ { x e E i "c(x)< [xe, A]}.
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Choose any y e E ~ E ' , a By Definition 3.1 and the finiteness of the truth value set there follows the existence of some Xl ~-E with
[ y e A ] = [ y = x l A 2XI~.A]. If xl ~E' again there exists s o m e x 2 E E with x2~xl and [ x l e A ] = [ x l - x 2 ^ 2 x2 e A]. It follows also that I-yl; A ] = [ - y - - x 2 ^ 2 x 2 E A ] ,
because by choice of xl, x2" [ y e A ] = E y = - - x l ^ 2x1 ~-x 2 A 2x2EA-!
<[Y--X2 ^ zx2z, A]
[ye, A]=[y=~Xk ^ 2 X k S A ] . This holds true for every y~ E~E'. Therefore (E', ~ ['E') is a generator of A, and by construction a canonical one. In the case of L~, obviously every fuzzy set with a finite generator has a basis. But it is an open problem, if each fuzzy set has at most one basis.
4. Set algebra again
First let us consider the problem to get a generator of the fuzzy union (intersection) of two fuzzy sets by the union (intersection) of their generators. To have shorter formulations, some more notation: For every subset E of E and every z : E---, W let be
and for every fuzzy set A let the function rA be given by ra (X)=[xgA] and have the maximal domain which is possible from context. Proposition 4.1. Assume that (El,zl ), (L2, r 2 ) a r e generators of A,B, and define jbr i = 1 , 2 z ~ ' E I u E 2 - * W by ~ PEi=zi and z~(x)=0 otherwise. Then ~ ( E 1 u E 2, z'1 v 11:~ )-- A u t B. Slf E~E ' is empty, already (E, 3) is canonical.
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Proof. For every fuzzy element
a :9
aeA wl B~qe.,x(a=x ^ 2TI(X)) V I:~E2X( a ~ X ^ • -~,:::]Etk.)EeX({a~x A 7Z'I(X)) V l ( a ~ x ^
2"if2(X)) A 2~'2(X)))
2, r (x)v
Corollary 4.2. Assume that E I , E 2 are i~enerating sets of A; B. Then A u I B = q ~ ( E 1 u E 2 , zA v IZB).
Proof. Put E=Et wE2" then for every fuzzy element a"
a e A w l E ~ 3 u x ( a = x ^ 2za(xi) v l--4ex(a=x ^ 2ZB(X)) --~'CA(a} V 1zB(aJ~a~Awl B, and hence the assertion follows. One cannot prove a corresponding result for c~t. Consider e.g. fuzzy elements a , b , c such that ½ < _ [ a - c ] , [ b - c ] < l and the fuzzy set A={a}r~l{b}. Obviously Ice.A]>½, but the gener~ting sets {a}, {b} of {a}, {b} are disjoint, and hence their intersection does not ,contain a generating set for A. Furthermore, if [ a = b ] = O , also {a}w {b} does not contain a generating set. for A. Clearly, always Ac
,B--q
(IA!uIBI, ,A ^ I "B ,
but generally here ]AI, [B} one cannot :replace by other generating sets of A. B. Therefore it is possible to define znother fuzzy intersection by a property like that proved in Proposition 4.1. To avoid dependence from choice of a special generator, let us consider only such fuzzy sets which have a unique basis. This property was proved for fuzzy sets with respect to Ln, n finite. Therefore, for the rest of this section we restrict ourselves to the finite-valued cases W = I4/.. For every fuzzy set A we denote by IIA!l the basic set of A.
Definition 4.3. For all A, B:
In the same manner one can define set algebraic operations using the connectives ^ 2, v 2. Also il, ll lloll ould have b ¢n Chosen as generating set for an intersection. We will not. Eiscuss the set algebraic properties of n~' only because a choice among all such possible definitions should be made by their use in applications. 9By abuse of language, we consider also r~(x) and r2(x) as constants for truth values.
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Proposition 4.4. For all A,B, (IIAllnIIBll, tA ^1 ts) is a basis for A n T B . Proof. By definition, IIAIIc~IIB]I is a generating set of A c~'~B. If it were not minimal, there, would exist some x, y e IIAII~',IIBII with tA(X) ^ l t s ( x ) < [ x - y
^ 2 ('CA(Y)
^ I
rB(y))].
We can assume ta(y) < tB(y). Hence tA(X ) A 1 ts(x)<--[x=-Y A 2TA(y)] <_tA(X).
The case tA(X)
Proposition 4.5. For fuzzy sets A, B consider the classical sets A/B = {x ~ ]]AI] J z,4(x ) > tB(X)},
AIIB= {x ~ IIAII n
IIBII I (x)-
Then a basic' set of,: ..,~B is ( A / B u B/A u AJJB). Proof. We put C = A / B w B / A u A [ [ B . generator of A u~B. Because of
Ilal( u (Inll = c ~ (IIAII~A/B) ~
First, let us prove that [C,r A v lts) is a
(IIBII\B/A),
we need only to prove the following two fuzzy inclusion relations: tP(IJAII~A/B,'CA v t t s ) - - c t P ( C , tA V l ZB),
(1)
Consider any X ~ IIAII'~A/'B" for (1) it is enough to have "CA{X) V t'Ca(X)<---[XBfp{C,'c A V 1 t B ) ] .
By xeIIA[I~A/B , tA(X)=
tdx)=< [ x - y ^ zt~(y)].
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S. Gottwald
Choose >,o llBI! with this property and with maximal value for *B(Yo). For every such Yo, *A(Yo)
x¢iiAll),
~A(X) V l'cB(x)='~B(X)<[x~.q~(C,r.a v , ~)]. This gives (1). (2) one proves in the same way. Second, (C, ~A V t ZB) is a minimal generating set. Otherwise there exists x, y E C with x :/: y and TA(X ) V I Z'B(X)<=[x=Y A 2('CA(y) V ITB(Y))]"
Clearly,
xeA[IB and yeAIIB"also not
and x,y~llall;
x , y ~ A / B (otherwise "ca(x)
a contradiction)and not
ZB(X)<.CA(X)<[x=Y^2ra(y)]
Proposition 4.6. For fuzzy sets A,B, C v~ith unique bases we have (i)
AnTB~Bn'~A,
(ii)
A n '~(B n '~C ) "~ (A n '~B ) ~ '~C ,
(iii)
A u t (An*B).~A,
(iv)
An'~(Au~B)c_A,
(v)
A~'~A~A,
(vi)
(A ut B)n~ (A u l C)~_A u1 (Bc,'~ C),
(vii)
A n T (Bu, C)~_ (AnT B ) u t (Ac~;C).
Proof. By straightforward computations. The given fuzzy inclusions one cannot improve to fuzzy equalities. But e.g. one has the classical implications
IIAII'-IIBII A T(A , B) A, IIAII IIBII A B- A TC B TC. 5. Fuzzy ordered pairs and cartesian products Definition 5.1. For fuzzy elements a, b let be
(a,b)=df{{a,b}, {b}}. These ordered pairs have the property that there ho!d
(a,b)~(c,d),-,.(a=c ^ lb=-d)v t ( a - d t t u - b ^ lb=-c)
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141
and furthermore also ( a , b ) ~ (c,d)¢~ a - c ^ , b = d .
But the full many valued version of this last classical equivalence does not hold. Obviously there hold a--c ^ lb-d--,(a,b)~.(c,d),
but the corresponding implication in the opposite direction fails, because the fuzzy implication a-d ^ ld-b
^ lb-c-,a-c
(*)
^ lb-d
does not hold. Therefore some properties of symmetry with respect to first and second components of (a,b) do not generalize from classical ordered pairs to these fuzzy ones. As a consequence some essential results of classical set theory would not generalize. In a slightly different context I have considered such a situation already in [5]. Here I'll choose another way to overcome this difficulty. The key remark is that (,) holds true if one chooses the element a in such a way that [ a = d ] ~ {0, 1 } for each fuzzy element d. This is the case if a is an urelement. Therefore we add to our system the assumption that there exist two different distinguished urelements ao and ao0.
Definition 5.2. For fuzzy elements a, b let be ( a , b ) = af{(ao, a), (.oo, b)} • For this modified ::2tion of fuzzy ordered pair now there hold (a,b) ~ (c,d).-~a=c ^ I b=d
and hence also (a, b) ..~ (c,d).--~(a, b) = ( c , d ) . We use it to define fuzzy cartesian products and later on also fuzzy relations and fuzzy mappings.
Definition 5.3. For fuzzy sets A, B let be
2x-=- (y, =511.
A
It is easy to see that always A x 2 B ~ A x lB. Fur'~hermore one proves by straightforward computation thai there hold 4 x , B ~ , U { { (,a,b)lt,,~A,,.b~R1
lae[AlandbelBI}.
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S. Gottwald
Proposition 5.4. (i) aeA A lbcB,--,
(ii)
a~A A 2bcB--,
Proof. Because of the remarks before this proposition we need only to prove part
( ~ ) of (i). Again by these remarks we have
[(a,b)v~A x t e l - - s u p [
< sup [(a=c A 2ce.A) ^ l(b=d A 2 d t ; B ) ] 1° d~ IBI
<[ae, A A lbF. B]. Item (ii) of Proposition 5.4 does not hold with ~ instead of --,. To see this, choose some truth value t with 0:ptq= 1 and fuzzy elements x,a such that rxe { a ] ] = t . In this case there hold
[(x,x)e. {a}× 2{a}] > [xl; {a} A 2Xe. {a}]. Nevertheless we still have
aeA A 2beB,~:,v.A x 2B. The full many valued version of this equi~ alence one can prove in the finite-valued case for such fuzzy ordered pairs , which belong to the basis of A x 2B. With the help of suitable tautologies of fuzzy propositional logic now one can prove e.g.
A x,B,~O~--,A.~Ov ,B~O, ( A u t B) x , C ~ ( A x , C ) u t (B x,C),
( A n t B ) x,C~_(A x,C)c~,(B x,C). It is easy to prove many other such elementary properties. We will be interested here only in proving monotony properties. Proposition 5.5. A c_ B A 2 C c_ D--, A x t C c_ B x 1 D. Proof. For any a, b we have
A ~_B~(ae A - , a e B) ~(ae, A A l b e . C ~ a e B A l b e C ) 1°Here we have used that (Pt A !P2) A , (p.~ A J P4)"'(PJ A 2P3) A I (P2 A 2P4) is a tautology of fuzzy propositional logic (cf. [8]).
Set theory f or fuzzy sets of higher level
143
and hence by Proposition 5.4 (i) also
A~_B--,((a,b)sA x t C ~ ( a , b ) e , B x 1C). Therefore there hold
A G B ~ A x , C G B x IC. and by the transitivity of _ also our statement. Proposition 5.6 [Ln, n finite]. A _ B - , A x 2 C G B x 2 C.
Proof. Consider only such fuzzy elements a, b for which (a, b) belongs to the basis of A x 2 C. Proceed as in the last proof, finally apply Proposition 3.5 (i). 11 Proposition 5.7. Suppose there hold 3x(x8 C). 12 Then also there hold true (i)
A~_B~--~A X
(ii)
A~B~--~A x 1C,,~,B x l C.
IC_____.B x 1 C ,
Proof. We prove only (i), because (ii) then is an easy corollary. By the previous results we have
A
x
1C~_B x ~C,--~VxVy(xsA A ~ysC ---,xsB A ~ye.C).
Consider some real number 6>0. There exists some element hi6) such that [b(6)8C]> 1 - 6 . Now one can estimate
] [ A ~ _ B ] - [ V x ( ( x , b ( 6 ) ) e A x ~ C ~ ( x , b ( f ) ) s B x , C)]] <6,(6), ][A x 1C-~B x t C ] - [ V x ( ( x , b ( b ) ) s A x ~ C - - , ( x , b ( 6 ) ) s B x iC)]1<32(6) such that lim 6, (6) = lim 52(6)'-0.
~i--,0
(i~O
Therefore there holds also
IimI[A~--B]--[A × ,C~_B × , C-! I = 0 , 6--.0
which means [ A G B ] = [ A x 1C~_B × ~C] and is equivalent to (i). ~tThis proof also works for L~, if A x 2 C has a canonical generator (E,r) with E a set of fuzzy ordered pairs and for every (a,b)eE, r((a,b))=[aeA A 2bcC]. 12By our conventions, this means [3x(xeC)] = 1, not only Icl nonempty.
144
S. Gottwald
Again we consider finally the problem to build up a generator of A x ,B from generators of A and/3.
Proposition 5.8 Assume that (Et,Tx),
generators of A,B. Define T: El x E 2 - , W by z((x,y>)=zl(X)A2z2(y) and put E = { < x , y ) l ( , x , y ) ~ E l x E 2 and z((x,y))--/:O}. Then (E,z ~'E) is a generator of A x 2B. (E2,1:2) are
Proof. For all fuzzy elements a, b we have
(a,b>e.A x 2B *--*31alu:tlBlr((a,b>=(u,v> ^ 2ue.A ^ 2ve B). Furthermore we have
ueA~"3E,x(u--x ^ ,r, tx)) and a corresponding property for v e B, also there hold
( a - u A 1 b - v ) A 2u=-x--~a=x ^ 1 ( b - v A 2u=-x) and hence 31,41!l((a.:-u A i b ~ F ; ) A 2u=---x)--~a=---x A l b=--v.
All together leads to
(a,b>eA x 2B--,3ezx3e2y31alv((a=x ^ 1 b - v ) A 2 v = y ^ 2~ (X) ^ 2z2(Y))
--*~v~x3Ezy((a--x ^ lb=--y) A 2ZI(X) A 2T2(y)). But for every x; zl(x)<__ExeA] and z2(x)<=ExeB'[, hence also
3f.,x~E2Y((a,b>--(x,Y> ^ 2zl(x) ^ 2 z 2 ( y ) ) ~ ( a , b > e A x 2B. These last two fuzzy implications prove our proposition. Without proof we note that for L 3 this generator (E,r ~'E) is a basis of A x 2B, provided that (E~,,1) and (Ez,*2) are bases. But for all other L,, n > 3 finite, this does not hold true. Proposition 5.8 is false for x 1. Choose e.g. a truth value s ~ W +, s :/: 1 and fuzzy elements a, b, c such that [a - c] = g= 1 - s. In this case [(c, b>e {a} x 1 {b},] = s ^ ~g~O.
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145
Yet on the other hand
[(c,b)e { ( a , b ) } , ] = [ ( c , b ) - ( a , b ) A 2s]=g A zs=O. Hence staj,{b} are generating sets of {a}, {b}, while {(a,/')} is not a generating set of {a} x , {b}. 13
6. Fuzzy mappings Definition 6.1. A generator (E, ~) of a fuzzy subset X of a fuzzy cartesian product A x ,B, i.e. X _ A x ,B, is called good iff each element of E is a fuzzy ordered pair. A fuzzy subset F~_A ×IB is called a fuzz), ' mapping or a fuzzy binary relation (with respect to A, B) iff F has a good generator. It is a routine matter to extend the notion of fuzzy ordered pair, and to define fuzzy ordered n-tuples by recursion. Then one has also the general notion of a fuzzy n-ary relation.
Definition 6.2. For any fuzzy mapping F we define domain and range by ~(f)--df{X]]3y-7_Z(X=--y A z(y, z ) e F)}, : ~ ( F ) = df{X ![ :-ly 3Z(X ~ Z A z ( y , z ) e F ) } .
Proposition 6.3. For every fuzzy mapping F there holt/ (i)
cJ(r).~{x]]3z((x,z)sF)~,
(ii)
~ ( F ) ~ {x][3y((y,x)eF)}.
Proof. (i) For fuzzy ordered pairs we have
x=y~(x,z)=(y,z) and therefore 3z((x, z) e.F)~xe. ~(,~:)
--*3y3z((x, Z)=-(y,z) A 2()',,7)gF)
~3y3z((x,z)sF), hence all these fuzzy implications are biimplications. For (ii) the same method works. 13For the finite-valued logics L, this last remark gives the possibility to define a further cartesian product.
146
S. Gottwald
For every fuzzy mapping F with respect to A, B there hold F ~ ~(F) x 19~(F),
9(F)c_A,
.~(F)~_B.
As a special case we have also
CjlA x ~B)~_A,
.~(A x tB)~_B.
But in both cases the fuzzy inclusion cannot be improved to a fuzzy equality, provided [BI or IAI is nonempty. For the first one an example is given by {a} x t{b}, for any s t W + with s:'41. However there hold
~y(3'eB)~(A x IB)~A, 3 x ( x e A ) ~ . ~ ( A x 1B)~B.
Proposition 6.4. For fuzz)' mappings F, G there hold (i)
F ~_G--,~(F)c_~(G),
(ii)
F ~ G--+~(F)~ ~(G).
Proof. It is enough to prove (i). Let E be a generating set of ~(FI, then one has using the existence of a good generator of F:
F ~_G-~Vx Vy((x, y ) e , F ~ ( x , y)e, G) -~ VEx Vy ( ~ F ~ xg @(G ) )
~ ¥ r x ( x e ~(F )-~xe ~(G ) ). Now apply Proposition 3.5 (i). Of cJurse the same statements hold true for the ranges of F, G.
Proposition 6.5. Let E be a good generating ~et of a fuzz)' mapping F. Put do(E) = { x l V r ( ( x , y ) e E ) } and rg(E)={ylVx((X,y)~E)}. Then d o l E ) i s a generating set of ~(F) and rg(E) a generating set of ~(F).
Proof. It is enough to consider ~(F). In this case for every fuzzy element a there hold
ae.~(Fj~bq~(x,y)((a=x A lb=y) A 2(x,y)eF) -~3r(x,y)(a==-x A 2 ( x , y ) e F ) --~3do(elx(a-----X A 2 ~rgtely((.v,, y ) e F ) )
~3do~E~X(a--x ^ 2 x e ~ ( F ) ) --' 3dole~x(ae, ~ (F ) ).
Set theoryfor fuzz)' sets of higher level
147
Hence all these fuzzy implications are fuzzy biimplications. Now apply Proposition 3.2 OiL Definition 6.6. For every fuzzy mapping F let be f -1 ---df{Xll3y3z(X.=--(y,Z) A 2 ( z , y ) e F ) } .
It is obvious that there hold Vx ~/y((x, y ) e F - 1*--~(y,x) e F).
Proposition 6.7. Suppose ~,..,z) to be a good generator of a .fuzz]' mapping F. Put E ' = l ( x , Y ) l ( Y , x ) ~ E } and define z " E ' ~ W + by r ' ( ( x , y ) ) = r ( ( y , x ) L Then (E',r') is a good generator or F - 1 Proof. Obvious.
As a consequence we note that there hold for fuzzy sets A, B (A x,B) - I ~ B x , A . For every fuzzy mapping F we have furthermore F . ~ ( F - 1 ) -1,
~(F- I)~,#(F),
.~(F -1 ) ~, CJ(F).
Is G a further fuzzy mapping we have also Fc_G,-~F-1 ~ G -1
9
F ~. G~_~,F - t .~ G -1"
Definition 6.8. For f u z z y mappings F, G let be G~,F=dr Ixll3Yqztx-(Y, : ) A 2 3 u ( ( y , u ) e F A,(U,z>eG)),.
As usual we have also in this case Go2FGG~,IF.
Proposition 6.9. For f u z z y mappings F, G there hold VaVb((a,b>e.G~,t F , ~ x ( ( a , x ) e . F
^ ](x,b)eG~).
S. Gottwaid
148
Proof. For fuzzy elements a, b there hold (a,b)eG :l F--+3y3z3x{(a-y ^ l b--z) A 2 ( ( y , x ) s F
Al(x,z>~6)) ~ 3 y ~ z 3 x ( ( a - y A 2(Y,X)e,F) A t ( b - z A 2(x,z)~G))
-,By3zqx((a,x)e.F A t ( x , b ) s G ) --*(a,b)sG :l F. This proof does not work for =2. And indeed Proposition 6.9 does not hold for G z F . To give a counterexample we choose some truth value s t W + with s<½ and fuzzy elements a,b,c,d such that [a=d]=[b=d]=[c=d]=s. We put F ={(c,d)~ and G= {(d,c) )). Now there hold [(a,d)~F]=[(d,b)e.G]=s and hence
[3x((a,x)r.F A 2(x,b>8 G)] =0. But on the other hand we have
[(a,b)e. G zF]=s=/=O because of [ ( c , c ) s G : 2 F ] = I and [(a, b> =- (c, c)] = s. Again it is possible to prove a lot of elementary properties for these notion,;. We mention only
Fc_H--,F , G ~ H ,G, IF: G)-I ~,G-I ~. F-1 F~I (G:t H)~ IF :l G)=I H. The proofs are without difficulties. As a consequence of Corollary 4.2 the fuzzy union F w l G of two fuzzy mappings F, G is itself a fuzzy mapping. On the other hand F ~, G is not always again a fuzzy mapping. Definition 6.10. Suppose that F, G are fuzzy mappings. Consider E = ~(a, b) l(a, b) ~ IG n,F[} and the function z" E--+ W + defined by
r((x, y>)= [(x, y> e G n,F]. Then by Gc~zF we mean that fuzzy mapping which has (E, z) as a generator This operation is in our theory the "right" intersection for fuzzy mappings.
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149
7. Fuzzy images Definition 7.1. For fuzzy mappings F and fuzzy sets A let be
F#~=af{xllSySz((y8A ^ , < y , z ) e F ) ^ 2x=zl}.
Proposition 7.2. Suppose that F is a fuzz)' mapping and A a fuzzy set with generator (E, r ). Then there hold
(i) (ii)
F'".~{:,:IISy(yeA ^,(y,x>eF)}, F ' 2 ' ( A ) ~ {x II 3ey(z(y) ^ 2 < y , x ) ~ F ) I .
Proof. Straightforward. Item (ii) of this proposition does not hold for F'~'(-). On the other hand only F ' ~ < . ) and the iteration of this kind of fuzzy image has the usual connection with one of the products of fuzzy mappings, introduced in the last .section.
Proposition 7.3. For fuzzy mappings F, G the'e hold (G~ 1F)~I~
Proof. Straightforward. A corresponding result which relates G :2 F and G ' 2 ' ( F . . . ) does not hold. As elementary properties one can prove e.g
Ft2~ ~ F ~ I ~ ( A ) _=~(F), F(°(A> ~O~--~An , ~ ( F ) ~ O ,
~(F)~_A ~ ( F ) . ~ F~I'(A). There hold also the following monotony progerties:
F ~_G ~F"'~- G'"
F ,~G~F°'(A)',eG"D(A), A ~ B-oFt'~(A) :~F°~(B). Finally we generalize the classical characterization of mappings by their images.
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150
Proposition 7.4. For fuzzy mappings F, G, there hold
(i)
F ~ G ~ V u X (F(2)(X)~G(2)(X)).
(ii)
F~,,G~-~V~IX (F(2)(X),~,G(2)(X)).
Proof. (ii) is a corollary of (i). Part (-~) of fi) is a direct consequence of the remarks immediately before this proposition. Therefore we have only to prove part ~ - ) of (i). Consider any a and the fuzzy set X = {a], then VMX (F(2)(X) ~ G ( 2 ) ( X ) ).--~
~VxI3y(y =-a ^ 2(y,x)e.F)--.3z(z-a ^ 2 ( z , x ) e G)) ~Vx(a-a A 2(a,x)eF .3z((a,x)=-(z,x) ^ 2(z,x)eG)) --,Vx((a,x)eF-o(a,x)e G). Now use that a was arbitrary and that F ".as a good generator. Then by Proposition 3.5 (i) the result follows. For the other kind of image we get also
F "-=G ~ Vu X (F(I)(X) ~ G(I)(X) ). But the corresponding fuzzy biimplication is not generally true. We can prove only a weaker version. Proposition 7.5. For Juzzy mappings F, G there hold
(i)
F~GoV~X(F(I)(X)c_G(!)(X)),
(ii)
F~,,G.~VMX(F(1)(X)'~G(I)(X)).
Proof. Again only part ( ~ ) of (i) is to prove. Hence assume F(I)(X)~G(t)(X) for every fuzzy set X. Now consider any fuzzy elements a,b and the truth value s =[(a,b)eF]. In this case we have [beF(1)({a}~)]>s, hence also [be, G(1)({a}s)]>s and therefore [(a,b)e,G]>s. Because F has a good generator this means [F ~_G] = 1 by Proposition 3.5 (i).
References [1] [2] [3J [4]
C.C. Chang, Algebraic analysis of many valued logics, Trans. Am. Math. Soc. 88 (1958) 467-490. B.R. Gaines, Foundations of fuzzy reasoning, Int. J. Man-Machine Studies 8 {1976) 623-668. R. Giles, Lukasiewicz logic and fuzzy set theory, Int. J. Man-Machine Studies 8 (1976) 313-327. J.A. ~oguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145-174.
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[5] S. Gottwald, Untersuchungen zur mehrwertigen Mengenlehre, Math. Nachrichten, Part I: 72 • (1976) 297-303, Part II: 74 (1976) 329-336, Part III: 79 (1977) 207-217.
[61 S. Gottwald, Ein kumulatives System mehrwertiger Mengen, Dissertation B, Leipzig {1975). [7] S. Gottwald, A cumulative system of fuzzy sets, in: Set Theory and Hierarchy Theory, Mem.
[8] [9] [1o] [11] [12] [13] [14] [15]
Tribute A. Mostowski, Bierutowice 1975, Lecture Notes in Mathematics 537 (Springer, Berlin, 1976) 109-i19. S. Gottwald. Theoretische Betrachtungea fiber Fuzzy-Logik, Schriftenreihe Weiterbildungszentrum Math. Kybernetik Rechentechnik, TU Dresden, Heft 27/77 (1978) 3-22. T.J. Jech, Lectures in aet Theory, Lecture Notes in ivlathematics 217 (Springer, Berlin, 1971 ). D. Klaua, ~ber einen zweiten Ansatz zur mehrwertlgen Mengenlehre, Monatsb. Deutsch. Akad. Wiss. Berlin 8 (1966) 161-177. D. Klaua, Grundbegriffe einer mehrwerdgen Mengenlehre, Monatsb. Deutsch. Akad. Wiss. Berlin 8 (1966) 781-802. J. g, ukasiewicz and A. Tarski, Untersuchungen fiber den Aussagenkalkfil, C. R. S~ances Soc. Sci. Lettr. Varsovie, cl. Ill 23 (1930) 3050. M. Mizumoto and K. Tanaka, Some properties of fuzzy sets of type 2, Information and Control 31 {1976) 312-340. C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis IBirkhauser. Basel, 1975). L.A. Zadeh, Fuzzy sets, Information and Control 8 (t965) 338353.
SET THEORY FOR F U Z Z Y SETS OF HIGHER LEVEL Siegfried G O T T W A L D Sektion Mathematik, Kari-Marx-Unit, ersitiit, 701 Leipzig, German Democratic Republic Received January 1978 Revised May 1978 We extend the usual notion of fuzzy set in such a way that the elements of fuzzy sets again can be fuzzy sets. For such fuzzy sets of higher level the fuzzy set theoretic operations are generalized up to the notion of a fuzzy mapping. In our presentation of the results we use a suitable many valued logic, indicating in this way the close formal connections between fuzzy and classical set theory.
1. Introduction Fuzzy sets are the mathematical models for extensions of vague notions. They are characterized by membership functions which permit grades of membership between full membership and non-membership. It is appropriate to consider these membership grades as truth values of a suitable many valued logic, and to develop the theory of fuzzy sets in the language of such a logic (cf. e.g. [2, 3]). We confine ourselves to the real numbers between zero and one, as was done by Zadeh [15] a n d since then in most papers on fuzzy sets. Generalizations are possible, but we will not consider them here. We use in this paper logics which often are called fuzzy logics. They are versions of the many valued logics of Lukasiewicz [12]. Some remarks on these logics are given in [8]. Their truth value sets are W, = [0, 1] and Wm= { k/m- 1]0 < k < m - 1 } for every natural number m > 21" their connectives -7 A 1, A 2, V t, V 2, -" ~-* are defined by" - l s = l - s , SAlt=min{s,t}, SA2t=max{O,s+t--1], s v t t - max {s, t}, s v 2t = min { l,s + t:,, s---,t = min { 1, 1 --s + t}, s~t = 1 --Is - t[. Of course, we have defined only their corresponding truth value functions, using always the same denotations for connective and function. Later on we use also ^ , or v ,, if in f;ome expression H both A 1, A 2 or both v 1, v 2 are permitted. If H laa,'~, more than one occurrence of the index ~, then it always has to be interpreted in the ~ame way. [ H I is the truth value of H. Bounded quantifiers V~, 3M, M a nonempty class of classical set theory, are defined by [VMxn(x) = inf {[n(a)][ae
[3 MxH(x)] = sup
M},
{[H (tl)]la e M ].
These logics we denote by Lm, m > 2, and L , " if only some unspecified of them is considered we use L resp. Wwithout index. By W ÷ we mean W~{0}. ~For m = 2 all is reduced to classical logic and classical set theory. 125
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S. Gottwald
Usually fuzzy sets are defined over some "universe of discourse"--which itself is a classical set. But not always such an universe of discourse can be a set of "wellknown" things. If one considers e.g. vague notio~is of higher order--i.e, properties of notions, relations between properties of notions etc., e.g. the vague notions "is a vague notion" itself--the corresponding fuzzy sets must be such that their "'elements" themselves are fuzzy sets. Therefore fuzzy sets of higher level are interesting objects to ~tudy. One can expect applications e.g. in linguistics (cf. [14, p. 40]). The technical problem posed by that discussion is the construction of an universe of discourse which is closed v.nder the formation of fuzzy subsets. We attack this problem by construction of a cumulative system of fuzzy sets. Such systems have already been described b3 Klaua [10, l l ] and the authoc [5]. But Klaua's system has the disadvantage that one is not able to develop it fully within the above mentioned many valued logic because of lack of a suitable fuzzy identity. The basic ideas for the present construction ,tre given i.n [7-1.2 Starting with a set R(0) of urelements, one defines simultaneously by transfinite recursion rank sets R(=) of fuzzy sets of level at most =, the fuzzy membership predicate e, a fuzzy rank relation i - , and fuzzy identity - . E = ~ =~0,fl(~) is the class of all fuzzy elements, M=E'~R(O) is the class of fiizzy sets. We use a,b,c,x,y,z,.., as variables for fuzzy elements and A,B,C,X, Y,Z,... as variables for fuzzy sets. Unbounded fuzzy quantification means bounded fuzzy quantification over E. The fuzzy rank relation has truth value one, [x i--y] = l, iff there does not exist an ordinal fl with yeR(flD and xCR(fl), i.e. iff the level of x is not greater than the level of y. Each R(~+ 1) is the union of R(a) and the set of all those functions .f : R(~)~[0, 1], for which for every fl<~. there exist x,7 with f l < 7 < ~ , x~R(7), and f (x) ~ 0, and for which for all x, y e R (~)
[x-y] A 2f(x)Sf(y). R(2), 2 limit ordinal, is the union of all R(fl), fl<2. Fuzzy membership is defined by (*)
[xeY]={~ (x) if x~d°(y)'otherwise"
with do(y) ~he domain of function y, doly) empty for y urelement. And fuzzy identity is in the case of x~R(O) or y~R(O): i_x=y]=f~
if x--),, otherwise,
and in the case of x, y ~ M:
[x-y]=[Vz(ze, x~-~z~y) A
2 x [ - y A 2Y[-- X].
2Both [7] and the present paper are part of the authors Habilitationsschrift 16].
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As an auxiliary notion let us introduce a f u z z y equality .~ for fuzzy sets by [A ~ B]=[Vx{xeA~--~xc, B)]. Then we have always 3 a -bA2be.A~aeA, a -b.--~Vx(aex~-~be..x), A .~ B,-.Vx(xe A~-~xr, B ), A - B , - - - , A , ~ B A 2 ( A ~ B A 2B[-- A). Our construction of the cumulative hierarchy of fuzzy sets has some analogies with the construction of Boolean valued models for (classical) set theory (cf. e.g. I-9, p. 53]). But there, the Boolean membership values have a much more complicated definition as our (,), the same is the case for Boolean identity" it is an open question if that way out will be helpful for the theory of fuzzy sets. We prefer (,) as we wish to identify the membership grades with truth values of fuzzy membership. Using the vocabulary of our logics L and r., =, E , ~ as binary relation constants, we can build up well-formed formulas in the usual way. If H(x) is such a wff with free variable x and with properties (i) for all fuzzy elements a, b there holds a-b
^ 2H{al-.H(b),
(ii) there exists an ordinal ~ such that for every fuzzy set bC~R(:~) the expression HIb) has truth value zero, then there exists exactly one fuzzy set A with property Vx(xc, A.-.H(x)).
{xllHtx)l
Let us writ,J for this fuzzy set A. As special fuzzy sets we consider the empty.litzzy set 0, characterized by [ x e 0 ] = 0 for every x e g ; fuzz)' singletons {al, for each truth value t and each fuzzy element a, characterized by [ x e { a ' j , ] = [ x - a ] ^ 2 t : and jit_-zy rank sets .~1~, characterized by Ix e ~/~] = 1 iff x ~ R{~) and [x e, .~#~] = 0 otherwise. We write {al for {a} 1" In [7] we define also ajitzzy inclusion relation ~_ by A ~_B.-~Vx(xe, A ~ x e , B ). heor ~ as a corollary we have A ~ B*-,A c_B ^ I Bc_A. 3If we state a fuzzy expression H in this way, this is short for the statement [H] = 1.
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With every fuzzy set A we correlate a classical set, its support
IAI = {x ~ E I I:xe A] 4:o}. By =~, ~ , A , V we denote implication, biimplication, generalization, and existential quantification of classical logic.
2. Elementary algebra of fuzzy sets Definition 2.1. For fuzzy sets A, B let
Au,B=dr{XllxaA v ~x~B}, An,B--df{Xllx~A A tx~B}. These definitions are correct ones, since for every y: y ~ x A 2 ( x g A *1 x e B )
~ ( y = x n 2Xe A ) * t ( y - x n 2Xe. B) ~ y e A "1 Ye B with , for ,\, v. But it is not l:..ssible to define in this way e.g a second conjunction with A 2 instead of A ~, because not always [y=--x A 2 { x c A A 2 x s B ) ] < [ y e A
A 2ysB].
Later on we will find another way to define corresponding operations for fuzzy sets by ^ 2, v 2. As an easy consequence of Definition 2.1 we note
AGB~AutB~B~AntB~A.
(*)
For the next results, it is convenient to change the formal definition of a lattice slightly, using in all defining equations (distributivity included) ~ instead of -=. Because for fuzzy sets A,B, A = B iff A ~ B--i.e. iff [A ~, B ] = 1--for systems of fuzzy sets the modified definition is equivalent to the usual one. Now we have
Proposition 2.2. For every ordinal ~ a distributive lattice with smallest and largest element is given by ['R(~ + 1 )~-R (0), n l , u t, O, ~d~].
Furthermore [M, "~t, wl,O]
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and also JR(2 )"~R (01, n , , u 1,0], for limit ordinals 2, are distributive lattices with smallest element.
Proof. By easy computations. From (,) now we see that the relation c_ is the right lattice ordering for these lattices. This gives us some monotony properties for n t, w l with respect to ~_, but we can prove more generally" A ~_B~Ac~I C~_Bc~I C, A ~_B---,A u t C ~_Bul C. As a corollary we have the full many valued substitutivity for fuzzy equality: A ~ B - - , A n t C ~ B n I C, A~.B~AuIC~Bu1C. ( o r r e s p o n d i n g results for - instead of ~ are true because for - there hold many valued versions of the well-known identity axioms of logic (cf[7]). One proves also A~_B A tA~_C~*A~_Bc~ IC, A~_C A IB~_C~-~AutB~_C. It is easy to extend these results to corresponding ones for a second fuzzy inclusion, defined by A~_*B*~,A~_B A 2 A F B . Furthermore, it is possible to prove yet more set algebraic fuzzy properties by using results of fuzzy propositional logic. We mention only the fuzzy implications A c_BoA ut
(BnIC)~.(AuIB)c~IC,
(AntB~Ac~tC) 2 A2Au1B~AutC~B~C" in the last expression with (H) 2 short for (H A2H), H the expression "Ac~tB ~An 1 Later on we need also xF" AutB*-'*xF'- A v l x V B, xF- AnlB.-*xV" A A lXF" B.
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Now, the notions of generalized union and generalized intersection shall be defined--however, we will do it not for fuzzy sets of fuzzy sets, but for systems, i.e. classical sets, of fuzzy sets. Definition 2.3. Let M, N be systems of fuzzy sets, N non-empty:
In both cases, ix IIH(x )} is a correct description of a fuzzy set; e.g. always there holds (Q for 3, vl
= - x A 2Qny(xs y ) ~ Q u y ( z - x
A 2XSy}
--,Qny(ze y). Clearly. always
Atu,B=U{A,B },
Am,B=~{A,B}.
For systems M, N of fuzzy sets and fuzzy sets A one can also prove e.g.
U(M,-,Nt UM,-,,UN, VuX(XcA}~UMc_A, M subsystem
of N =~U M c_ L_)N ;
for nonempty systems M, N one has also
VMX(A~_X)--)A~_(~M, M subsystem of N=#,(~N ~_N M . More details are proved in [6]. As remarked in [7], for certain A the fuzzy predicate "x~_ A" does not define a fuzzy set. Hence there we defined the.~zzy power set by* PA=,jf{x[ix ~*A}.
Proposition 2.4. (i) A ~_* B,-,PA c_PB, (ii)
PALJtPBGP(A~IB),
(iiit
P(Ac~IB)GPAnlPB.
4For urelements x we put [x_ *A] =0.
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Proof. (i) By transitivity of __, for every x A~_B~(x~_A~x~_B).
Assuming A E B, it is easy to get (x~_A---,x~_B)~(x ~_* A--.x ~_* B).
Both fuzzy implications give A ~_*B--*PA~_PB.
The reverse fuzzy implication is obvious. For 0i) one can use the fuzzy implication x~_A v l x G B ~ x ~ _ A u ~ B ,
and for (iii) t h corresponding one, x~_An~ B--,x~_A ^ ~x~_B.
Corollary 2.5. (i) A E (i'.')
B,-,PA E PB,
PA ~ PB,-,PA c_ * PB.
Proof. (i) By definition of the fuzzy rank relation we have 5 PAEPB*~
A
V (XEY),
X~IPA I YEIPBI
and also PA E P B ~ A E B because of A~IPA I and B ~ I P B I. Furthermore AE B ~
/~ (X E B)=:,PAE PB. X~IPAi
Hence A E B'~PA E PB,
and this classical biimplication is equivalent to the fuzzy one, for :X E Y can have only truth values zero and one. (ii) one proves using (i) and Proposition 2.4 (i). SRemember footnote (3), and hence that "PA E PB¢.,.'' "" is short for "IPA E PB] = 1 ~ " "'.
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In classical set theory, Proposition 2.4 (iii) is an equation. In the fuzzy case one cannot prove the second inclusion, needed for equality, because only [ P A n 1 P B ~_P{A n l B)] 4=0
(,)
is true. Clearly, if [X e P A n ~ P B ] = 1 for some X, X ~_*A and X ~*B, hence X * A n~ B. Therefore {,). But for any tlath value r > 0 one can choose three fuzzy elements a, b, c of equal rank with
[ a - b ] = [b-=c] = [a--c] =0" put for Y={a}" A~=Yw,{Y},., B~=Yu,{{b}}~, and X = Y u , { { c } } , . Now A , n ~ B r = Y and hence IX ~_*A,]=[X ~ * B , ] = l - r . On the other hand [X E A~n~ B , ] = 0 and therefore IX ~_* A,c,, B,] =0. All together this gives [PAr n l P B , ~ P(A,n I B~)] < r. Hence (,) is sharp. 3. Generators of fuzzy sets
It is an essential property of our fuzzy sets of higher level that there exist fuzzy singletons A={al=:xllx-al with card Ihl>l. Hence need not know all elements of the support IAI or a fuzzy set A for a description of A, but only a suitable subset of them. Definition 3.1. A generator of a fuzzy set A is an ordered pair (E, r) of a subset E of IAI and a function r" E ~ W + such that A=
~
E}.
A set E is a generating set of A iff there exists a function z such that (E, z) is a generator of A. Proposition 3.2. (i)(E,x) :s a generator of A iff z "E--, W ~ and
/~ i [ a e A ] = s u p ( [ a - x ] a
.~E
^ 2fix))).
E
(ii) A set E is a generaling set of A iff there hold
Va(as A*--~:iEx(a==-x ^ 2xaA)).
(,)
Proof. (i) is obvious. (if) If E is a generating set of A, consider any generator (E, z) of A. For in this case always z ( x ) < [ x e A ] for x e E and also [ a = x ^ 2 x e A ] <[aeA], (,) is an easy consequence of (i). Now suppose (,). Define z ' E ~ W + by z(x)=[xeA] and apply (i) to (E,z).
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133
Corollary 3.3. Assume that E is a set with property
Ax
^ 2yra]).
yeE
Then E is a generating set of A and A 4=0.
Proof. By Proposition 3.2 (ii).
Proposition 3.4.
Suppose that EI,E 2 are generating sets of the filzzy sets A I,A 2.
Then there hold A 1 [-'- A2~""~VEIX 3E2Y(X[-" y).
Proof. Straightforward from Definition 3.1 and the construction of the fuzzy sets.
Proposition 3.5. (i) (ii)
Let E l , E 2 be generating sets for A I , A 2 . There hold
A 1 ~A2*-...}VEtX(XI5 A l ~ x 8 A2) , A l .~ A2~"-*VEI,.,E2X(X8 A,,~-.,xe Az).
Proof. (ii) is an easy corollary of (i). The fuzzy implication Al GA2-}VF.,x{xsAI'-*xsA2)
is obvious. The fact that there hold x e A 2 A 2y=--X-"*ysA2
for any x, y and therefore xsA2-..~ysA 2 v 2y~x
holds too, proves (xgAI~xgA2).-o{xgA,-.*ysA2
v
2y~x).
Hence there hold VEIX(xeAI"'*xsA2)~(3F.~X()'=-x
^ 2Xg A1)~yr; A2).
Now we apply Proposition 3.2 (it) and fuzzy generalization with respect to 3' and get VE,X(X8 A 1 -"*x15 A2)--*A 1c_ A2"
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Corollary 3.6. For fuzzy elements a, b,c,d and truth values s, t there hold 6
{a}~___ {b}, ,-~(a=-b ^ 2t) v 2 ~, Oil
{a,b}.,~_ {c,d}, , - . ( ( ( a - c v , a ~ d ) ^ ~( b - c v ~b - d ) ) ^ 2 t ) v
2 ~.
Proof. By straightforward computations. We apply this result to some more special cases. The case s = t yields e.g. the following simplification: {a}~_ {o}~ ,--~a=b v ,~--~{a}~.v, {bJ~. Later on we are also interested in: {a,b}c{c,d}~--~(a=-c v 1 a - d ) A , ( b = c v ,b=-d),
{a,b} ..~ {c,d}.'.-,(a-c A~ b = d ) v ~(a=d ^ ,b=c). Finally we mention the following coincidences of fuzzy identity and fuzzy equality:
{al {a,b} ~{c,d},-~{a,b}=_{c,d}.
Definition 3.7. Suppose (E,r) as a generator of A. We call (E,r) a canonical generator of A iff A x ~ E ( r ( x ) = [ x c A ] ) ; and,we call (E,~)a minimal generator or a basis of A iff E is a minimal generating set of A with ~espect to classical inclusion. Proposition 3.8. Every basis of a fuzzy set A is also a canonical generator of A. Proof. Let (E, 3) be a basis of A, and assume that for some Xo ~ E there hold Z(Xo) < [ x o e A ] . Consider E ' - E ~ { x o } and z ' = z [ ' E ' . Because there hold [xoeA] =supy~E max {0, [Xo = y] + z ( y ) - 1} it exists some Yo e E with property r(Xo) < max {0, [Xo - Yo] + z ( y o ) - 1} and hence with {Xo},~xo~{Yo},tyn~. Therefore also (E',~') is a generator of A, contradicting the minimality of E. Hence (E, z) is canonical. For every fuzzy set A the support IA[ is a generating set of A; if one defines z'[A[-~W + by z ( x ) = [ x e A ] , then (IAl, ),s a canonical generator of A. Hence every fuzzy set has a canonical generator. But does there exist always a basis? The answer is affirmative for the finite-valued cases and regative for the infinitevalued one. 6To have short formulations, we use "s", "t" also as constants of our language, g is short for "-Is.
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T h e o r e m 3.9. [L~]. There exists a fi~zzy set without any basis.
Proof. We choose a sequence (r,),,~n of truth values, 7 which is monotone decreasing and converges to zero. We choose also some b ~ E with OF- b. Let be A = ~{{{b},.} 1 _ , . I n , N } . Suppose, E is a basic set of A. E must be nonempty and every m ~ N. Because for every n < m ~ N,
:#{{b j~rmJ"~ for
E{b}, - { t , } . . ] = we have for every k ~ N
Take atE. Assume [ a e A ] < l . Consider E'=E'~{a}, choose k ~ N > [ a ¢ A ] ; obviously, for every n > k and B=(,.J{{c}t~,ajlc~E', ---
with 1 - - r k
~°
[{b},..~; B] = 1 - r,, = [{b}, ~;A], hence A=B, and E is not minimal. Thus [ a e A ] = 1 for every aeE. Also, if a,a' eE, ~ ~ have limit 1 Hence a = a ' the sequences ( [ a = { b j,,]),,~ and ([a'={bj~,,]),,~ and E is a singleton: E = {aj. For this a, lim,_. [ a = { b j~j = l , hence 01-a. But already lirn,~, [x~. {bi~,] = 1 has x = 0 as only solution. Thus no such a exists, A has no basis. Proposition 3.10. [L,, n finite]. For each nonempty family (E:),~! oJ" generating sets
of a fuzzy set A also their intersection ~ I E ~ is a generating set of A Proof. Put E*=(']i~tEi. If A = 0 then always Ei is empty, and hence also E*. In this case E* is a generating set of A. Furthermore suppose A 4:0. Then our proof is finished if for E* there hold the supposition.,_, of Corollary 3.3. Therefore choose any fuzzy element a. 'We have to show the existence of some v e E* with
[a~A]<[a--y n 2ye, A] ~. Case 1" a ~E*. There is nothing to prove" take y = a . Case 2: a~Ui~lEi~E *. There exists j ~ l with a~Ej, and k1~I with aCEk,. Furthermore, there exists x~ ~ Ek, with |
[as A] = m a x {O,[a=xl]+rk,(Xl)-- 1,,
°
first, according to Proposition 3.2. (i) the value [a8 A] is given as a supremum, but this supremum here has to be a maximum. Hence we have
[ae, A]<[a=-xl n 2xtaA]<[xIIzA]. 7N is the set of natural
numbers.
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We are done if x t ~ E*. Otherwise there exist
k2 E I
with x~ ¢ Ek2, and x2~Ek2 with
[Xlt~A]'<[Xt--x 2 A 2 x 2 8 A ] < [ x 2 8 A ] and with
[ a s A ] = [ a = x l A 2Xt =--X2 ^ 2 x 2 8 A ] < [ x = x 2
A 2x28A] •
The last equality holds true, for we have
[a=--xl
A 2Xl ~ x 2 A 2 x 2 ~ , A ] < [ a = x l
A
2xt~,A]<[a~,A]
by the properties of = and e, and by the choice of xt, x2 also
[ae, A ] < [ a = x l A 2 x t r . A ] < [ a = x l /', 2xl =x2 ^
2x28A].
Anew we are done if x2 ~ E*. Otherwise this procedure continues, and produces a sequence x~, x2, x3 .... with
Ix, ~ A] <[x2e A] < [x3e A] < " " which therefore must be finite, if the procedure stops with Xk, then Xk ~ E* and for y-~ Xk the desired property holds. Case 3: a~[_)i,~Ei. There exist j e l and z~_Ej with [aeA]<[a=--z A 2 z e A ] . By case 2 there exists some y e E * with [ z e A ] < [ z = y A 2ysA]. Hence this y has also in this case the desired property.
Theorem 3.11 [L,, n finite]. Every fuzzy set has exactly one basis. Proof. Consider the system 9Jl of all generating sets of a given fuzz) set A. 9J/is nonempty and (']gJie~JJl by Proposition 3.10. Hence (']gJl is the only minimal generating set of A. In addition to the existence of bases another interesting question concerns the existence of canonical generators. We have seen that each fuzzy set has a canonical generator But does each generator (E, z) of a fuzzy set A contain a canonical generator--in the sense that for some subset E' of E the pair (E',r I'E') is a canonical generator of A? Again the answer is affirmative for t h e finite-valued cases and negative for the infinite-valued one. For L ~ we cot~sider the fuzzy set A which was constructed in the proof of Theorem 3.9. Put t r ( x , ) = l - r n for every l l < n ~ N . It is easy to see that ({x,] 1 < n ~ N l , a ) i s a generator of A, not containing any canonical generator.
Propositiq,n 3.12 [L,,, n finite]. For every generator (E,r) of a fuzzy set A there exists a stibset E' of E such that (E', z ~E') is a canonical generator of A. Proof. Let a generator (E, z) of A be given, and consider the subset
E ' = L ~ { x e E i "c(x)< [xe, A]}.
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137
Choose any y e E ~ E ' , a By Definition 3.1 and the finiteness of the truth value set there follows the existence of some Xl ~-E with
[ y e A ] = [ y = x l A 2XI~.A]. If xl ~E' again there exists s o m e x 2 E E with x2~xl and [ x l e A ] = [ x l - x 2 ^ 2 x2 e A]. It follows also that I-yl; A ] = [ - y - - x 2 ^ 2 x 2 E A ] ,
because by choice of xl, x2" [ y e A ] = E y = - - x l ^ 2x1 ~-x 2 A 2x2EA-!
<[Y--X2 ^ zx2z, A]
[ye, A]=[y=~Xk ^ 2 X k S A ] . This holds true for every y~ E~E'. Therefore (E', ~ ['E') is a generator of A, and by construction a canonical one. In the case of L~, obviously every fuzzy set with a finite generator has a basis. But it is an open problem, if each fuzzy set has at most one basis.
4. Set algebra again
First let us consider the problem to get a generator of the fuzzy union (intersection) of two fuzzy sets by the union (intersection) of their generators. To have shorter formulations, some more notation: For every subset E of E and every z : E---, W let be
and for every fuzzy set A let the function rA be given by ra (X)=[xgA] and have the maximal domain which is possible from context. Proposition 4.1. Assume that (El,zl ), (L2, r 2 ) a r e generators of A,B, and define jbr i = 1 , 2 z ~ ' E I u E 2 - * W by ~ PEi=zi and z~(x)=0 otherwise. Then ~ ( E 1 u E 2, z'1 v 11:~ )-- A u t B. Slf E~E ' is empty, already (E, 3) is canonical.
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Proof. For every fuzzy element
a :9
aeA wl B~qe.,x(a=x ^ 2TI(X)) V I:~E2X( a ~ X ^ • -~,:::]Etk.)EeX({a~x A 7Z'I(X)) V l ( a ~ x ^
2"if2(X)) A 2~'2(X)))
2, r (x)v
Corollary 4.2. Assume that E I , E 2 are i~enerating sets of A; B. Then A u I B = q ~ ( E 1 u E 2 , zA v IZB).
Proof. Put E=Et wE2" then for every fuzzy element a"
a e A w l E ~ 3 u x ( a = x ^ 2za(xi) v l--4ex(a=x ^ 2ZB(X)) --~'CA(a} V 1zB(aJ~a~Awl B, and hence the assertion follows. One cannot prove a corresponding result for c~t. Consider e.g. fuzzy elements a , b , c such that ½ < _ [ a - c ] , [ b - c ] < l and the fuzzy set A={a}r~l{b}. Obviously Ice.A]>½, but the gener~ting sets {a}, {b} of {a}, {b} are disjoint, and hence their intersection does not ,contain a generating set for A. Furthermore, if [ a = b ] = O , also {a}w {b} does not contain a generating set. for A. Clearly, always Ac
,B--q
(IA!uIBI, ,A ^ I "B ,
but generally here ]AI, [B} one cannot :replace by other generating sets of A. B. Therefore it is possible to define znother fuzzy intersection by a property like that proved in Proposition 4.1. To avoid dependence from choice of a special generator, let us consider only such fuzzy sets which have a unique basis. This property was proved for fuzzy sets with respect to Ln, n finite. Therefore, for the rest of this section we restrict ourselves to the finite-valued cases W = I4/.. For every fuzzy set A we denote by IIA!l the basic set of A.
Definition 4.3. For all A, B:
In the same manner one can define set algebraic operations using the connectives ^ 2, v 2. Also il, ll lloll ould have b ¢n Chosen as generating set for an intersection. We will not. Eiscuss the set algebraic properties of n~' only because a choice among all such possible definitions should be made by their use in applications. 9By abuse of language, we consider also r~(x) and r2(x) as constants for truth values.
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Proposition 4.4. For all A,B, (IIAllnIIBll, tA ^1 ts) is a basis for A n T B . Proof. By definition, IIAIIc~IIB]I is a generating set of A c~'~B. If it were not minimal, there, would exist some x, y e IIAII~',IIBII with tA(X) ^ l t s ( x ) < [ x - y
^ 2 ('CA(Y)
^ I
rB(y))].
We can assume ta(y) < tB(y). Hence tA(X ) A 1 ts(x)<--[x=-Y A 2TA(y)] <_tA(X).
The case tA(X)
Proposition 4.5. For fuzzy sets A, B consider the classical sets A/B = {x ~ ]]AI] J z,4(x ) > tB(X)},
AIIB= {x ~ IIAII n
IIBII I (x)-
Then a basic' set of,: ..,~B is ( A / B u B/A u AJJB). Proof. We put C = A / B w B / A u A [ [ B . generator of A u~B. Because of
Ilal( u (Inll = c ~ (IIAII~A/B) ~
First, let us prove that [C,r A v lts) is a
(IIBII\B/A),
we need only to prove the following two fuzzy inclusion relations: tP(IJAII~A/B,'CA v t t s ) - - c t P ( C , tA V l ZB),
(1)
Consider any X ~ IIAII'~A/'B" for (1) it is enough to have "CA{X) V t'Ca(X)<---[XBfp{C,'c A V 1 t B ) ] .
By xeIIA[I~A/B , tA(X)=
tdx)=< [ x - y ^ zt~(y)].
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S. Gottwald
Choose >,o llBI! with this property and with maximal value for *B(Yo). For every such Yo, *A(Yo)
x¢iiAll),
~A(X) V l'cB(x)='~B(X)<[x~.q~(C,r.a v , ~)]. This gives (1). (2) one proves in the same way. Second, (C, ~A V t ZB) is a minimal generating set. Otherwise there exists x, y E C with x :/: y and TA(X ) V I Z'B(X)<=[x=Y A 2('CA(y) V ITB(Y))]"
Clearly,
xeA[IB and yeAIIB"also not
and x,y~llall;
x , y ~ A / B (otherwise "ca(x)
a contradiction)and not
ZB(X)<.CA(X)<[x=Y^2ra(y)]
Proposition 4.6. For fuzzy sets A,B, C v~ith unique bases we have (i)
AnTB~Bn'~A,
(ii)
A n '~(B n '~C ) "~ (A n '~B ) ~ '~C ,
(iii)
A u t (An*B).~A,
(iv)
An'~(Au~B)c_A,
(v)
A~'~A~A,
(vi)
(A ut B)n~ (A u l C)~_A u1 (Bc,'~ C),
(vii)
A n T (Bu, C)~_ (AnT B ) u t (Ac~;C).
Proof. By straightforward computations. The given fuzzy inclusions one cannot improve to fuzzy equalities. But e.g. one has the classical implications
IIAII'-IIBII A T(A , B) A, IIAII IIBII A B- A TC B TC. 5. Fuzzy ordered pairs and cartesian products Definition 5.1. For fuzzy elements a, b let be
(a,b)=df{{a,b}, {b}}. These ordered pairs have the property that there ho!d
(a,b)~(c,d),-,.(a=c ^ lb=-d)v t ( a - d t t u - b ^ lb=-c)
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and furthermore also ( a , b ) ~ (c,d)¢~ a - c ^ , b = d .
But the full many valued version of this last classical equivalence does not hold. Obviously there hold a--c ^ lb-d--,(a,b)~.(c,d),
but the corresponding implication in the opposite direction fails, because the fuzzy implication a-d ^ ld-b
^ lb-c-,a-c
(*)
^ lb-d
does not hold. Therefore some properties of symmetry with respect to first and second components of (a,b) do not generalize from classical ordered pairs to these fuzzy ones. As a consequence some essential results of classical set theory would not generalize. In a slightly different context I have considered such a situation already in [5]. Here I'll choose another way to overcome this difficulty. The key remark is that (,) holds true if one chooses the element a in such a way that [ a = d ] ~ {0, 1 } for each fuzzy element d. This is the case if a is an urelement. Therefore we add to our system the assumption that there exist two different distinguished urelements ao and ao0.
Definition 5.2. For fuzzy elements a, b let be ( a , b ) = af{(ao, a), (.oo, b)} • For this modified ::2tion of fuzzy ordered pair now there hold (a,b) ~ (c,d).-~a=c ^ I b=d
and hence also (a, b) ..~ (c,d).--~(a, b) = ( c , d ) . We use it to define fuzzy cartesian products and later on also fuzzy relations and fuzzy mappings.
Definition 5.3. For fuzzy sets A, B let be
2x-=- (y, =511.
A
It is easy to see that always A x 2 B ~ A x lB. Fur'~hermore one proves by straightforward computation thai there hold 4 x , B ~ , U { { (,a,b)lt,,~A,,.b~R1
lae[AlandbelBI}.
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Proposition 5.4. (i) aeA A lbcB,--,
(ii)
a~A A 2bcB--,
Proof. Because of the remarks before this proposition we need only to prove part
( ~ ) of (i). Again by these remarks we have
[(a,b)v~A x t e l - - s u p [
< sup [(a=c A 2ce.A) ^ l(b=d A 2 d t ; B ) ] 1° d~ IBI
<[ae, A A lbF. B]. Item (ii) of Proposition 5.4 does not hold with ~ instead of --,. To see this, choose some truth value t with 0:ptq= 1 and fuzzy elements x,a such that rxe { a ] ] = t . In this case there hold
[(x,x)e. {a}× 2{a}] > [xl; {a} A 2Xe. {a}]. Nevertheless we still have
aeA A 2beB,~:,
A x,B,~O~--,A.~Ov ,B~O, ( A u t B) x , C ~ ( A x , C ) u t (B x,C),
( A n t B ) x,C~_(A x,C)c~,(B x,C). It is easy to prove many other such elementary properties. We will be interested here only in proving monotony properties. Proposition 5.5. A c_ B A 2 C c_ D--, A x t C c_ B x 1 D. Proof. For any a, b we have
A ~_B~(ae A - , a e B) ~(ae, A A l b e . C ~ a e B A l b e C ) 1°Here we have used that (Pt A !P2) A , (p.~ A J P4)"'(PJ A 2P3) A I (P2 A 2P4) is a tautology of fuzzy propositional logic (cf. [8]).
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and hence by Proposition 5.4 (i) also
A~_B--,((a,b)sA x t C ~ ( a , b ) e , B x 1C). Therefore there hold
A G B ~ A x , C G B x IC. and by the transitivity of _ also our statement. Proposition 5.6 [Ln, n finite]. A _ B - , A x 2 C G B x 2 C.
Proof. Consider only such fuzzy elements a, b for which (a, b) belongs to the basis of A x 2 C. Proceed as in the last proof, finally apply Proposition 3.5 (i). 11 Proposition 5.7. Suppose there hold 3x(x8 C). 12 Then also there hold true (i)
A~_B~--~A X
(ii)
A~B~--~A x 1C,,~,B x l C.
IC_____.B x 1 C ,
Proof. We prove only (i), because (ii) then is an easy corollary. By the previous results we have
A
x
1C~_B x ~C,--~VxVy(xsA A ~ysC ---,xsB A ~ye.C).
Consider some real number 6>0. There exists some element hi6) such that [b(6)8C]> 1 - 6 . Now one can estimate
] [ A ~ _ B ] - [ V x ( ( x , b ( 6 ) ) e A x ~ C ~ ( x , b ( f ) ) s B x , C)]] <6,(6), ][A x 1C-~B x t C ] - [ V x ( ( x , b ( b ) ) s A x ~ C - - , ( x , b ( 6 ) ) s B x iC)]1<32(6) such that lim 6, (6) = lim 52(6)'-0.
~i--,0
(i~O
Therefore there holds also
IimI[A~--B]--[A × ,C~_B × , C-! I = 0 , 6--.0
which means [ A G B ] = [ A x 1C~_B × ~C] and is equivalent to (i). ~tThis proof also works for L~, if A x 2 C has a canonical generator (E,r) with E a set of fuzzy ordered pairs and for every (a,b)eE, r((a,b))=[aeA A 2bcC]. 12By our conventions, this means [3x(xeC)] = 1, not only Icl nonempty.
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Again we consider finally the problem to build up a generator of A x ,B from generators of A and/3.
Proposition 5.8 Assume that (Et,Tx),
generators of A,B. Define T: El x E 2 - , W by z((x,y>)=zl(X)A2z2(y) and put E = { < x , y ) l ( , x , y ) ~ E l x E 2 and z((x,y))--/:O}. Then (E,z ~'E) is a generator of A x 2B. (E2,1:2) are
Proof. For all fuzzy elements a, b we have
(a,b>e.A x 2B *--*31alu:tlBlr((a,b>=(u,v> ^ 2ue.A ^ 2ve B). Furthermore we have
ueA~"3E,x(u--x ^ ,r, tx)) and a corresponding property for v e B, also there hold
( a - u A 1 b - v ) A 2u=-x--~a=x ^ 1 ( b - v A 2u=-x) and hence 31,41!l((a.:-u A i b ~ F ; ) A 2u=---x)--~a=---x A l b=--v.
All together leads to
(a,b>eA x 2B--,3ezx3e2y31alv((a=x ^ 1 b - v ) A 2 v = y ^ 2~ (X) ^ 2z2(Y))
--*~v~x3Ezy((a--x ^ lb=--y) A 2ZI(X) A 2T2(y)). But for every x; zl(x)<__ExeA] and z2(x)<=ExeB'[, hence also
3f.,x~E2Y((a,b>--(x,Y> ^ 2zl(x) ^ 2 z 2 ( y ) ) ~ ( a , b > e A x 2B. These last two fuzzy implications prove our proposition. Without proof we note that for L 3 this generator (E,r ~'E) is a basis of A x 2B, provided that (E~,,1) and (Ez,*2) are bases. But for all other L,, n > 3 finite, this does not hold true. Proposition 5.8 is false for x 1. Choose e.g. a truth value s ~ W +, s :/: 1 and fuzzy elements a, b, c such that [a - c] = g= 1 - s. In this case [(c, b>e {a} x 1 {b},] = s ^ ~g~O.
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Yet on the other hand
[(c,b)e { ( a , b ) } , ] = [ ( c , b ) - ( a , b ) A 2s]=g A zs=O. Hence staj,{b} are generating sets of {a}, {b}, while {(a,/')} is not a generating set of {a} x , {b}. 13
6. Fuzzy mappings Definition 6.1. A generator (E, ~) of a fuzzy subset X of a fuzzy cartesian product A x ,B, i.e. X _ A x ,B, is called good iff each element of E is a fuzzy ordered pair. A fuzzy subset F~_A ×IB is called a fuzz), ' mapping or a fuzzy binary relation (with respect to A, B) iff F has a good generator. It is a routine matter to extend the notion of fuzzy ordered pair, and to define fuzzy ordered n-tuples by recursion. Then one has also the general notion of a fuzzy n-ary relation.
Definition 6.2. For any fuzzy mapping F we define domain and range by ~(f)--df{X]]3y-7_Z(X=--y A z(y, z ) e F)}, : ~ ( F ) = df{X ![ :-ly 3Z(X ~ Z A z ( y , z ) e F ) } .
Proposition 6.3. For every fuzzy mapping F there holt/ (i)
cJ(r).~{x]]3z((x,z)sF)~,
(ii)
~ ( F ) ~ {x][3y((y,x)eF)}.
Proof. (i) For fuzzy ordered pairs we have
x=y~(x,z)=(y,z) and therefore 3z((x, z) e.F)~xe. ~(,~:)
--*3y3z((x, Z)=-(y,z) A 2()',,7)gF)
~3y3z((x,z)sF), hence all these fuzzy implications are biimplications. For (ii) the same method works. 13For the finite-valued logics L, this last remark gives the possibility to define a further cartesian product.
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For every fuzzy mapping F with respect to A, B there hold F ~ ~(F) x 19~(F),
9(F)c_A,
.~(F)~_B.
As a special case we have also
CjlA x ~B)~_A,
.~(A x tB)~_B.
But in both cases the fuzzy inclusion cannot be improved to a fuzzy equality, provided [BI or IAI is nonempty. For the first one an example is given by {a} x t{b}, for any s t W + with s:'41. However there hold
~y(3'eB)~(A x IB)~A, 3 x ( x e A ) ~ . ~ ( A x 1B)~B.
Proposition 6.4. For fuzz)' mappings F, G there hold (i)
F ~_G--,~(F)c_~(G),
(ii)
F ~ G--+~(F)~ ~(G).
Proof. It is enough to prove (i). Let E be a generating set of ~(FI, then one has using the existence of a good generator of F:
F ~_G-~Vx Vy((x, y ) e , F ~ ( x , y)e, G) -~ VEx Vy (
~ ¥ r x ( x e ~(F )-~xe ~(G ) ). Now apply Proposition 3.5 (i). Of cJurse the same statements hold true for the ranges of F, G.
Proposition 6.5. Let E be a good generating ~et of a fuzz)' mapping F. Put do(E) = { x l V r ( ( x , y ) e E ) } and rg(E)={ylVx((X,y)~E)}. Then d o l E ) i s a generating set of ~(F) and rg(E) a generating set of ~(F).
Proof. It is enough to consider ~(F). In this case for every fuzzy element a there hold
ae.~(Fj~bq~(x,y)((a=x A lb=y) A 2(x,y)eF) -~3r(x,y)(a==-x A 2 ( x , y ) e F ) --~3do(elx(a-----X A 2 ~rgtely((.v,, y ) e F ) )
~3do~E~X(a--x ^ 2 x e ~ ( F ) ) --' 3dole~x(ae, ~ (F ) ).
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147
Hence all these fuzzy implications are fuzzy biimplications. Now apply Proposition 3.2 OiL Definition 6.6. For every fuzzy mapping F let be f -1 ---df{Xll3y3z(X.=--(y,Z) A 2 ( z , y ) e F ) } .
It is obvious that there hold Vx ~/y((x, y ) e F - 1*--~(y,x) e F).
Proposition 6.7. Suppose ~,..,z) to be a good generator of a .fuzz]' mapping F. Put E ' = l ( x , Y ) l ( Y , x ) ~ E } and define z " E ' ~ W + by r ' ( ( x , y ) ) = r ( ( y , x ) L Then (E',r') is a good generator or F - 1 Proof. Obvious.
As a consequence we note that there hold for fuzzy sets A, B (A x,B) - I ~ B x , A . For every fuzzy mapping F we have furthermore F . ~ ( F - 1 ) -1,
~(F- I)~,#(F),
.~(F -1 ) ~, CJ(F).
Is G a further fuzzy mapping we have also Fc_G,-~F-1 ~ G -1
9
F ~. G~_~,F - t .~ G -1"
Definition 6.8. For f u z z y mappings F, G let be G~,F=dr Ixll3Yqztx-(Y, : ) A 2 3 u ( ( y , u ) e F A,(U,z>eG)),.
As usual we have also in this case Go2FGG~,IF.
Proposition 6.9. For f u z z y mappings F, G there hold VaVb((a,b>e.G~,t F , ~ x ( ( a , x ) e . F
^ ](x,b)eG~).
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148
Proof. For fuzzy elements a, b there hold (a,b)eG :l F--+3y3z3x{(a-y ^ l b--z) A 2 ( ( y , x ) s F
Al(x,z>~6)) ~ 3 y ~ z 3 x ( ( a - y A 2(Y,X)e,F) A t ( b - z A 2(x,z)~G))
-,By3zqx((a,x)e.F A t ( x , b ) s G ) --*(a,b)sG :l F. This proof does not work for =2. And indeed Proposition 6.9 does not hold for G z F . To give a counterexample we choose some truth value s t W + with s<½ and fuzzy elements a,b,c,d such that [a=d]=[b=d]=[c=d]=s. We put F ={(c,d)~ and G= {(d,c) )). Now there hold [(a,d)~F]=[(d,b)e.G]=s and hence
[3x((a,x)r.F A 2(x,b>8 G)] =0. But on the other hand we have
[(a,b)e. G zF]=s=/=O because of [ ( c , c ) s G : 2 F ] = I and [(a, b> =- (c, c)] = s. Again it is possible to prove a lot of elementary properties for these notion,;. We mention only
Fc_H--,F , G ~ H ,G, IF: G)-I ~,G-I ~. F-1 F~I (G:t H)~ IF :l G)=I H. The proofs are without difficulties. As a consequence of Corollary 4.2 the fuzzy union F w l G of two fuzzy mappings F, G is itself a fuzzy mapping. On the other hand F ~, G is not always again a fuzzy mapping. Definition 6.10. Suppose that F, G are fuzzy mappings. Consider E = ~(a, b) l(a, b) ~ IG n,F[} and the function z" E--+ W + defined by
r((x, y>)= [(x, y> e G n,F]. Then by Gc~zF we mean that fuzzy mapping which has (E, z) as a generator This operation is in our theory the "right" intersection for fuzzy mappings.
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149
7. Fuzzy images Definition 7.1. For fuzzy mappings F and fuzzy sets A let be
F#~=af{xllSySz((y8A ^ , < y , z ) e F ) ^ 2x=zl}.
Proposition 7.2. Suppose that F is a fuzz)' mapping and A a fuzzy set with generator (E, r ). Then there hold
(i) (ii)
F'".~{:,:IISy(yeA ^,(y,x>eF)}, F ' 2 ' ( A ) ~ {x II 3ey(z(y) ^ 2 < y , x ) ~ F ) I .
Proof. Straightforward. Item (ii) of this proposition does not hold for F'~'(-). On the other hand only F ' ~ < . ) and the iteration of this kind of fuzzy image has the usual connection with one of the products of fuzzy mappings, introduced in the last .section.
Proposition 7.3. For fuzzy mappings F, G the'e hold (G~ 1F)~I~
Proof. Straightforward. A corresponding result which relates G :2 F and G ' 2 ' ( F . . . ) does not hold. As elementary properties one can prove e.g
Ft2~ ~ F ~ I ~ ( A ) _=~(F), F(°(A> ~O~--~An , ~ ( F ) ~ O ,
~(F)~_A ~ ( F ) . ~ F~I'(A). There hold also the following monotony progerties:
F ~_G ~F"'~- G'"
F ,~G~F°'(A)',eG"D(A), A ~ B-oFt'~(A) :~F°~(B). Finally we generalize the classical characterization of mappings by their images.
S. Gottwald
150
Proposition 7.4. For fuzzy mappings F, G, there hold
(i)
F ~ G ~ V u X (F(2)(X)~G(2)(X)).
(ii)
F~,,G~-~V~IX (F(2)(X),~,G(2)(X)).
Proof. (ii) is a corollary of (i). Part (-~) of fi) is a direct consequence of the remarks immediately before this proposition. Therefore we have only to prove part ~ - ) of (i). Consider any a and the fuzzy set X = {a], then VMX (F(2)(X) ~ G ( 2 ) ( X ) ).--~
~VxI3y(y =-a ^ 2(y,x)e.F)--.3z(z-a ^ 2 ( z , x ) e G)) ~Vx(a-a A 2(a,x)eF .3z((a,x)=-(z,x) ^ 2(z,x)eG)) --,Vx((a,x)eF-o(a,x)e G). Now use that a was arbitrary and that F ".as a good generator. Then by Proposition 3.5 (i) the result follows. For the other kind of image we get also
F "-=G ~ Vu X (F(I)(X) ~ G(I)(X) ). But the corresponding fuzzy biimplication is not generally true. We can prove only a weaker version. Proposition 7.5. For Juzzy mappings F, G there hold
(i)
F~GoV~X(F(I)(X)c_G(!)(X)),
(ii)
F~,,G.~VMX(F(1)(X)'~G(I)(X)).
Proof. Again only part ( ~ ) of (i) is to prove. Hence assume F(I)(X)~G(t)(X) for every fuzzy set X. Now consider any fuzzy elements a,b and the truth value s =[(a,b)eF]. In this case we have [beF(1)({a}~)]>s, hence also [be, G(1)({a}s)]>s and therefore [(a,b)e,G]>s. Because F has a good generator this means [F ~_G] = 1 by Proposition 3.5 (i).
References [1] [2] [3J [4]
C.C. Chang, Algebraic analysis of many valued logics, Trans. Am. Math. Soc. 88 (1958) 467-490. B.R. Gaines, Foundations of fuzzy reasoning, Int. J. Man-Machine Studies 8 {1976) 623-668. R. Giles, Lukasiewicz logic and fuzzy set theory, Int. J. Man-Machine Studies 8 (1976) 313-327. J.A. ~oguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145-174.
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[5] S. Gottwald, Untersuchungen zur mehrwertigen Mengenlehre, Math. Nachrichten, Part I: 72 • (1976) 297-303, Part II: 74 (1976) 329-336, Part III: 79 (1977) 207-217.
[61 S. Gottwald, Ein kumulatives System mehrwertiger Mengen, Dissertation B, Leipzig {1975). [7] S. Gottwald, A cumulative system of fuzzy sets, in: Set Theory and Hierarchy Theory, Mem.
[8] [9] [1o] [11] [12] [13] [14] [15]
Tribute A. Mostowski, Bierutowice 1975, Lecture Notes in Mathematics 537 (Springer, Berlin, 1976) 109-i19. S. Gottwald. Theoretische Betrachtungea fiber Fuzzy-Logik, Schriftenreihe Weiterbildungszentrum Math. Kybernetik Rechentechnik, TU Dresden, Heft 27/77 (1978) 3-22. T.J. Jech, Lectures in aet Theory, Lecture Notes in ivlathematics 217 (Springer, Berlin, 1971 ). D. Klaua, ~ber einen zweiten Ansatz zur mehrwertlgen Mengenlehre, Monatsb. Deutsch. Akad. Wiss. Berlin 8 (1966) 161-177. D. Klaua, Grundbegriffe einer mehrwerdgen Mengenlehre, Monatsb. Deutsch. Akad. Wiss. Berlin 8 (1966) 781-802. J. g, ukasiewicz and A. Tarski, Untersuchungen fiber den Aussagenkalkfil, C. R. S~ances Soc. Sci. Lettr. Varsovie, cl. Ill 23 (1930) 3050. M. Mizumoto and K. Tanaka, Some properties of fuzzy sets of type 2, Information and Control 31 {1976) 312-340. C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis IBirkhauser. Basel, 1975). L.A. Zadeh, Fuzzy sets, Information and Control 8 (t965) 338353.