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A natural interpretation of fuzzy set theoryMamoru SHIMODA

Shimonoseki City UniversityShimonoseki, Japanmamoru-s@shimonoseki-cu .ax: jpAbstractWe present a new and natural interpretation offuzzy set theory in a cumulative Heyting valuedmodel for intuitionistic set thoery. By this interpre-tation we can consistently get various notions andproperties of fuzzy sets, fuzzy relations and fuzzymappings. We can consider notions such as oper-ations of fuzzy subsets of different universes, fuzzyrelations and mappings between fuzzy subsets.As far as fuzzy sets and fuzzy relations are consid-ered as extensions of crisp sets and relations, thisinterpretation seems o be most natural.

1. IntroductionIn ordinary fuzzy set theory, fuzzy sets axe identi-fied with mappings into the unit interval [0, 11 ofreal numbers, and the propeties or operations offuzzy seta and relations are defined by equations orinequalities([17,18]).Various definitions have beenproposed on the basic operations such as intersec-tion, union, complement of fuzzy sets, and compo-sition and inverse of fuzzy relations etc. (cf. e.g.,13, 4,5, 6, 7,8 1)). Sometimes only the definingequations are presented without explanation.We interpret fuzzy sets and fuzzy relations in themodel V H ntroduced by the author [12],a Heyt-ing valued model for intuitionistic set thoery, whereN is a complete Heyting algebra and is consideredthe set of truth values in the model. V H s a kindof so-called sheaf model, cumulatively constructedby transfinite iteration of power sheaf constructionover H. n the interpretation the canonical embed-ding from the class of all crisp sets into the modelplays an important role. Though the model is sim-ilar to the sheaf models in [9,15,16]and so on, ourinterpretation is original and unique.In the model we can easily define basic notions andoperations of sets and relations. By the canoni-n d embedding we can obtain most of the standarddefining equations of basic notions and operationsof fuzzy sets and relations. It shows that min, max

and intuitionistic negation are the most natural ba-sic set operations, and max-min composition is themost natural operation for composition of relations.We distinguish a generalized fuzzy set,a fuzzy sub-set of a crisp set (called a universe for the fuzzysubset), and a membership function of a fuzzy seton a crisp set (universe). There is a natural corre-spondence between fuzzy subsets of a crisp set andmappings from the crisp set to H, which preservesinclusion and basic set operations.We can naturally define basic set operations offuzzy subsets of different universes and define in-clusion relation between them. Thus we obtain ageneral theory of fuzzy sets where fuzzy subsets ofdifferent universes are treated in a natural and uniform way. We can also define relations and map-pings between fuzzy subsets of crisp sets, and givecharacterization of defining properties of equiva-lence relation, partial order, and linear order.There have been proposed several definitions offuzzy mappings or fuzzy functions, one of whichidentifies fuzzy mapping with fuzzy relation ([l, 2,4,7,8, 10,111). Our interpretation of fuzzy map-pings seems to be unique and quite natural. Theextension principle of Zadeh is expressed as a the-orem on images of mappings for natural extensionsof mappings between crisp sets.More details can be seen in [13,141, where the ex-tention of fuzzy set operations discussed in Subsec-tion 3.2 s not treated.2. The Heyting valued model2.1. Basic set constructionsLet H be a complete Heyting algebra with oper-ations and constants A , V, A V, +, 0 ,1, nd I.We construct an H-valued model for the extendedintuitionistic set theory, which is a first order intu-itionistic logic with predicates E,= and E (calledan existence predicate) together with axioms of in-tuitionistic set theory. LetV be the class of dl crispsets and On be the class of all ordinals.

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Definition 2.0.(1) The composit ion S o R is defined by:Let R, S E V H .D S 0 R ) = {(zz);X , z E V,"},E ( S 0 R ) = E R A E S ,S o R :(zz)c lEly(sRy A ySz II,

where Q is an ordinal satisfying 'DRU IDS E V, .( 2 ) The inverse relat ion R-' is deh ed by:

D(R- ' ) = { ( z y ) ; z , y V,"}, E (R - ') = E R ,where Q is a suitable ordinal satisfying D R C V z .

R-' : 4-- lY Il,

The definitions are not affected by the choice of theordinal a . If R and S re relations from U to II andfrom v to w respectively, then S o R is a relationfrom U to 20 and R-' is a relation from v to U .Deflnition 2.7.in V*.1) For A E V H , he image R ( A ) is defined by:

Let R be a relation from U to v

D ( R ( A ) )= Dv,R ( A ) : t3 1 3zE A)zRyll.E ( R ( A ) )= E RA E A ,

(2) For B E V H , he inverse image R- ' (B) isdefined by:D ( R - ' ( B ) ) = Du,R - ' (B ) :z c-) 1 3y E B)zRyll .

E ( R - ' (B ) ) = E R A E B ,

A relation R from U to v in Vw s called total if( Vz E u ) (3y E v) zRy) s valid, and is surjectiveif (Vg E v) 3z E u ) ( z R y ) s valid. R is injective if(Vz, E u)(Vz E v)(zRzA yRz -P = y ) is valid,and if (Vz E u ) (V y , z E v ) ( z R g A zRz P y = z )is valid, R is called univalent. Then R is total 8I, C R-' 0 R , R is surjective i I, C R o R-', Ris injective iffR-' o R E ,, and R is univalent iffR o R-' C I,.A relation R on U (that is, a relation from U to U ) iscalled wflemke if (Vz E u)(z&) is valid. SimilarlyR is symmetr ic i VxVy(xRy 3 &) is valid, istnrrnsitiue if VxVyVz(zRyA y R z + zRz) is valid, isa n t i s y " e t r i c i f V z V g ( z R y ~ y R z 3 2 = y) is valid,and is connected if ( V z E u)(VyE ~ ) ( z R yU ) isvalid. Then R is reflexive 8 , E R, is symmetric8 -' C R i f fR-' kr R , is transitive iff R o R C R,is antisymmetric 8 nR-' E I,, and is connectedi U x U - U R"l.An equivalence relation is a reflexive, symmetric,and transitive relation, an order relation is a reflex-ive, antisymmetric, and transitive relation, and ah e a r o d er is a connected order relation. Hencea relation R on U is an equivalence relation iffI 5 R , R-' E R , and R o R E R . If R is a

relation R-' o R is symmetric, and in addition if 2is total R-' o R is reflexive, and if it is univalentR-' o R is transitive.2.3. Mappings in the modelA mapping in V H s a total and univalent relationin V H . or U ,U, E V H , iscalled a mapping f romU to v i n V H f f is a total and univalent relationfrom U to v in v H . We write f :U an V H ff is a mapping from U to v in V H .Hence for everyf in V H , : in V H ff

f C U x U , I , E f-' o f , and f o f-' I,.We often write f(z)= y instead of ( z y ) E f or zfy.An in ject ion (resp. a su jec t ion) is an injective(resp. surjective) mapping, and a bi ject ion is aninjective and surjective mapping. A composition ofinjections (resp. surjections or bijections) is alsoan injection (resp. a surjection or a bijection).For a relation f from U to v , f is an injection iffI, f i : f-' o f and f o f-' E I,, is a surjection iffI, E f-'of and fof-' R. I,, and f is a bijection iffI, f i : f-' 0 f and f of-' M I,. Hence for a mappingf :U i n v H , f i s injective iff I, R. f - l o fand is surjective iff f o f-' R. I,. For a mapping fin V H , f-' o f is an equivalence relation.We can define images and inverse images of m a ppings similarly o those of relations.Most of the basic properties of relations and map-pings of usual set theory also hold in the model.3. The natural interpretation3.1. Natural interpretation of fuzzy s e t sIn the following, let X,Y, be crisp sets.Deflnition 3.1.set. For an H-fuzzy set A , the mappingEvery set in V H s an H-fuzzy

PA = :x ; 2 c-)115 E Allis called the membership function of A on X .An H-fuzzy subset of X is a subset of X in V H .Obviously A is an H-fuzzy subset of X iff A C xfor every A E V H .Lemma 3.1. Every mapping from X o H is themembership function of some H-fuzzy subset of X .Hence there is a natural correspondence betweenH-fuzzy subsets of X and mappings from X to H.The natural correspondence preserves inclusion (or-der) and the basic set operations.

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Definition 3.2.1) p(2) The mappings P A Y , p V u , p\u, y p : X + Ham defined by:

Let p , U :X .U iff p ( z ) 5 Y ( Z ) for all zE X.

p A U :z (z)A ~(z),p \ ~ : ~ t ) p ( ~ ) A - u ( z ) ,p v U :zc (2) v U@ ,- p :5 c-) 9p(z).

Theorem 1.members hip func tions below be on X .(1) I f A and B are H-fuzzy subsets of X I then

Let A,B be H-fuzzy sets and all the

A E B iff P A 5 p B , andA B iff P A = p ~ .

(2) PAnB = P A A CIS, P A U S = P A v pS1PA\B = P A \ P B , PW\A = ? P A .

3.2. Extension of fuzzy set operationsNow we consider the relation between H-fuzzy setsand mappings from different crisp sets to H.For p : X , efine p r y : Y by:

p [Y s called the restr ict ion of p on Y. ObviouslyDefinition 3.3. For p :X H, U : Y - H,the relations p 5 U and p 11 U are defined as ollows.IL tx = P a d P t y ) z = P W n2).

P V - S VtX,P H U p d u a n d u i p .

The relation 5 is a preorder compatible with theequivalence relation N If X = Y, then p 5 Ubecomes p 5 and p N u becomes p = Y .L e m m a 3.2. Let p :X and u :Y .

Now we extend Definition 3.2(2) and define opera-tions of mappings from diflerent sets t o H.

Definition 3.4. H , Y :Y ,themappings p A u , p V u , p \ u o n X U Y = Z a r edefined as follows.For p :X

c A = (P 2)A Yz),c v v = ( P tZ) v :UtZ),c \ = ( P tZ) \ /a

x e Y = (x Y) u (Y x .Lemma 3.4. Let p :X -+H,v Y -+HI and(1) p Y : z c r {2) p v v : z *

Proposition 3.5.u : Y +H.

p ( z ) A U ( Z ) ( Z E x nY)0 Z E X ) .P ( 4 ( 2 E x \ Y)p(z ) Y ( Z ) ( Z E x nY)44 z E Y \X .P ( 2 ) ( 2 E x \ Y )p ( z ) A ~ z ) Z E x nY)z E Y X .Assume tha t p :X and

p A U (resp. p V U) works as the i nhi mum (resp.supremum) of p and U in th e set of equivalenceclasses with respect to NTheorem 2. Let A,B be sets an Vw, ~ be themembership function of A on X , and p~ be themembership functio n of o n Y.(1) A n x E B n y i f l p ~ p E , and

A n X - B n p i f l p ~ ~ B .(2) I n addition, assume that A and B are H-fuzzysubsets o f X and Y respectively. Then

A C B i f f A 5 P S , andA - B i f l p A N- p ~ .

Theorem 3. Assume A, B are H-fuzzy ubsetsof X,Y respectively, and P A , C B are the member-ship functions on X and o n Y respectively. Letp ~ m ,A L J B ,A\B be the membership functions ofA n B ,AU B,A\ B on X U Y reappeetively.(2) PAUB = P A V P S .(3) PA\B = P A \ P E -1) pAnB = P A A P E -

Hence there is a natural correpondence between H-fuzzy subsets of meren t crisp sets and mappingsfrom those crisp sets to H, which preserves orderand operations.

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3.3. Relations between fuzzy subsetsEvery relation in V H s called an H-fvz zy relation.Definition 3.5. For R,u,v E V H ,R is an H -fuzzy relation from U to v if it is a relation fromU to v in V H . For every R E V H , he member-ship function of R f rom X to Y is the membershipfunction of R on X x Y, hence it is the mapping

PR :x y ; ( x y )c-) I(zY) E RII.An H-fuzzy relation from X to Y is an H - f u z z ysubset of X x Y .If A, B are H-fuzzy subsets of X, Y respectively,then obviously an H-fuzzy relation from A to B isan H-fuzzy relation from X to Y . For every set Rin V H , is an H - f u z z y relation from A to B iff

p R b y ) 5 P A (2) p B (y) for all 2 E x, E y,where PA, p ~ ,~ be the membership functions onX , Y , X x Y respectively.Proposition 3.6. Let A, B ,C be H-fuzzy subsetsof X , Y , Z respectively, R be an H-fuzzy relationfrom A to B , S be an H-fuz zy relation from B t oC, and P R , p s , P S ~ R , R - I e the membership fun c-t ions on X x Y,Y x 2, x Z , Y x X respectively.(1) The composition S R is an H-fuzzy relationfrom A to C , and for all x E X , z E Z

pSoR(XZ) = v P R ( 2 Y ) A PS(Yz)).Y Y

(2) The inverse relation R-' i s an H -f uz q re lat ion@m B to A, and for all x E X , y E Yp R - l ( / z ) = pR(XY).

These are extensions of the defining equations ofmax-min composition and those of inverse for ordi-nary fuzzy relations.Proposition 3.7. Let A, B be H -f uz zy subsets ofX ,Y respectively, R be an H-fu zzy relation from At o B, and p~ be i ts membership function on X x Y .( 1 ) For every C E V H , he image R(C) s a subsetof B an V H , nd for all y E Y

p R ( C ) ( Y ) = v ( P C ( z ) A P R ( X Y ) ) ,2EX

where p c and ~ R ( c ) re the membership functionson X and on Y respectively.(2) For every D E V H , he inverse image R-'(D)is a subset of A in V H , nd fo r all x E X

PR-I(D)(X)= V ( P D ( Y )A I L R ( X Y ) )where p~ and ~ R - I D ) are the membership func -tions o n Y and on X respectively.

V E Y

Proposition 3.8. Let A, B be H-fuzzy subsets ofX ,Y respectively, R be an H-fuzzy relation from At o B , and PA, p ~ g ,~ be their mem bersh ip unc tionson X , Y ,X x Y respectively.(1) R i s total ifl V y E Y p ~ ( ~ y )~ A ( x )or all2 E x.(2) R is suject ive i v S E x p ~ ( x y ) ~ B ( Y )o ra l l y E Y .(3) R is injective i f fp ~ x z ) ~ R ( Y Z ) 0 impliesx = y f o r a l l x , y E X and z E Y .(4) R is univalent i f f R ( X Y ) ~ R ( x z ) 0 impliesy = z f o r a l l x E X a n d y , t E Y .Theorem 4. Let A be an H-fuz zy subset o f X ,R be an H- fuz zy relation on A, and p a , p~ be theirmembership functions on X ,X x X respectively.( 1 ) R is reflexive i f f p ~ ( ~ 5 )~ A ( x )or all x E X .(2) R is s y m m e t r i c i f l p ~ ( z y ) ~ R ( Y X )o r a112, E x.( 3 ) R is tmnsitive i f l p ~ ( t y ) ~ R ( Y Z ) p ~ ( Z z )fo r al l x, ,z E X .(4) R i s antisymmetric i f f or all x, E X,p ~ ( 2 y ) p ~ ( y ~ )0 implies x = y .( 5 ) R i s connected i f f or all 5 , y E X,

pR(XY) v PR(YX) = P A ( X ) A p A ( 9 ) -If A = X and H = [0,1],he conditions are almostsame to the usual definitions in f uzzy literature, butthe condition ( 5 ) and the definition of linear orderis stronger than ordinary definitions.3.4. Fuzzy mappings and extension principleEvery mapping in V H s called an H-fuzzy mapping.Definition 3.6. An H-fuzzy mapping from U to vis a mapping from U to v in VH. n H-f i zzy map-ping from x to Y is a mapping from X to P inVH.Obviously an H - f u z z y mapping from U to v is atotal and univalent H-fuzzy relation from U to U.Theorem 5. Let A, B be H-fuzzy subsets of X , Yrespectively, and PA, p s be their membership func-tion on X , Y respectively. Suppose f E V H and1c = pf i s its membership fvnctaon o n X x Y . T h e nf is an H-fuzzy mapping fi-om A t o B i f $ satisfiesthe following three conditions:(R) (xY) 5 PA ( z ) A P B ( Y ) fo r al l 2 E x,(E)v $'(zY) = p A ( X ) fo r all 2 E x, E y

YY(U) (xy) A $(zz) > 0 implies y = zfor all x E X , and y, E Y.

Conversely, i f a crisp mapping : X x Ysatisfies the three conditions, there i s a n H - f u z qmapping f from A t o B whose membership functiono n X x Y is identical with .

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Proposition 3.9. Let A, B be H-fuzzy subsets ofX , Y respectively, f be an H-fuz zy mapping from At o E , and p f be its membership fun ctio n on X x Y .(1) For every C E V H , he image f C ) s a subsetof B i n V H , nd for ally E YPf(C)(Y)= v ( P C ( 4 A Pf 4),

where p c and pf c) are the membership functionson X and o n Y respectively.(2) For every D E V H , he inverse image f- (D)i s a subset of A in V H , nd for all x E X

Pf-1(&) = v (PdY) A Pf (SY)),

[5] J. Fodor and R. R. Yager, Fuzzy set-theoreticoperators and quantifiers, Fundamentals off i z zy se ts , Kluwer Academic Publishers, Dor-drecht, 2000, pp 125-193.[6] .Gottwald, F'uzzy uniqueness of fuzzy map-pings, Fuzzy Sets and Systems, Vol. 3, 1980,

=EX pp 49-74.[7] . Gottwald, Fuzzy Sets and fizzy Logic:Foundations of Appl imt ions - from a Math-ematical Point of View, Vieweg, Wiesbaden,1993.[SI S. Gottwald, Fundamentals of fuzzy rela-

DEY tion calculus, Fuzzy Modelling: Paradigmswhere p~ and Pf-l(D) are the membership f i nc -tions on Y and on X respectively.

and Pmctice, Kluwer Academic Publishers,Boston, 1996, p 25-47.[9] R. J. Grayson, Heyting-valued models for in-tuitionistic set theory, Applications of sheaves(Lecture Notes in Math. 753),Springer, Berlin,1979,pp 402414.[lo]W. . Nemitz, Fuzzy relations and fuzzyfunctions, &zy Sets and Systems, Vol. 19,[ll] S.V.Ovchinnikov, Structure of fuzzy binaryrelations, f i z z y Se t s and Syst ems , Vol. 6,1981, p 169-195.

valued models for intuitionistic set theory,

Theorem 6. Let cp :X --+ Y be a crisp mapping.Then the check set 8 i s a n H - f i z z y m a pp in g fromX to Y , and the followings hold.(1) For every A E V H , he image $ ( A ) i s a n H -uzzy subset of Y , and f or all y E Y(A)(y) = v 1986,pp 177-191.

2EX9(=)=vwhere PA andon X and on Y respectively.i s an H-fuzzy subset of X , and for all x E X

are the membership functions(2)For every B E V H , he inverse image @-'(B) [12]M. b o d a , Categorical aspects of Heyting-

Comment. Math. Univ. Sancti Pauli , Vol. 30,1981, D 17-35._ _where p~ and p p i ~ )n? the m embership func-tions on Y and on X respectively. [13]M. Shmoda, A natural interpretation offuzzy sets and fuzzy relations, submitted,(1) shows that the extension principle of Zadehholds.References[l] S. S. L. Chang and L. A. Zadeh, On fuzzymapping and control, IEEE h n s . on Sys-

tems, Man, and Cybernetics, Vol. 2, 1972,pp30-34.[2]M.Demirci, Fuzzy functions and their funda-mental properties, h z y ets and Systems Vol.106, 1999,pp 239-246.[3]D.Dubois, W.Ostasiewicz, and H. Prade,

F'uzzy sets: history and basic notions, f in -damentals of f i z zy se t s , Kluwer AcademicPublishers,Dordrecht, 2000,pp 21-124.[4]D. Dubois and H. Prade, y ets andSystems: Theory and Applicatio~zs,AcademicPress,New York, 1980.

1998.[14]M. Shimoda, A natural interpretation offuzzy mappings, submitted, 2000.[15] G. Takeuti and S.Titani, Heyting valued universes of intuitionistic set theory, Logic Sym-posium Hakone 1979, 1980 (Lecture Notes inMath. 891), Springer, Berlin, 1981, pp 192-306.[16]G. Takeuti and S. Titad, Fuzzy logic andfuzzy set theory, Arch. Math. ogic, Vol. 32,[17]L. A. Zadeh, Fuzzy sets, I n f o m a t i o n a nd

Control, Vol. 8, 1965, p 338-353.[18]L. A. Zadeh, Similarity relations and fuzzyorderings, Information Sciences, Vol. 3, 1971,

1992,pp 1-32.

pp 177-200.

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