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Volume 11, 2004

Some properties of mappings in the model V^H

Gokhan Çuvalcıoğlu

University of Mersin,

Faculity of Arts and Science

Department of Mathematics

e-mail: gcuvalcioglu@mersin.edu.tr

23 August 2003

Abstract: In this paper we study the basic decomposition theorem for mappings in V^H that is the model of intuitionistic set theory which is an extension of fuzzy set theory

Keywords: intuitionistic set, intuitionistic relation, intuitionistic mappings.

1. Introduction:

The notion of a fuzzy set was first introduced in 1965 by Zadeh [5]. The theory has found wideranging applications in such diverse fields as automata, control theory, decision theory, and social behavior pattern studies, to name just a few. At the same time, a fuzzy relation between X and Y as a fuzzy subset of X´Y was also proposet by Zadeh [5].

In [5], the fuzzy sets was introduced as an extension of crisp (usual) sets, the usual two-valued sets in ordinary set theory, by enlarging the truth value set to the real unit interval [0,1]. Fuzzy sets are characterized by mappings called membership functions into [0,1]. The basic fuzzy set operations (union, intersection, complement, Cartesian product, etc.) are defined by some equations or inequalities between the membership functions.

Intuitionistic fuzzy sets constitute a generalisation of the notion of a fuzzy set and were introduced by Atanassov [1].

The model V^H called Heyting model for intuitionistic set theory, where H is a complete Heyting algebra and is considered as the set of truth values in the model. The fuzzy sets and fuzzy relation was interpretted by shimoda [3]. It was indicated that V^H is a model for intuitionistic set theory by Titani [4].

In this paper, H is Heyting valued set with the classical operations, Ù, Ú, Ù, Ú, ®, Ø, 0, 1, £ .

In intuitionistic logic, the strictness axioms: xÎy®ExÙEy and x=y®ExÙEy. The theory of intuitionistic logic with existence predicate is developed in Scott [2].

Let V be the class of all usual sets and O_n be the class of ordinals. The model V^H is constructed as follows.

Definition 1: For every aÎO_n,

= Æ

=

= {u = <ïuï,Eu> : ïuï: Du®H, Du Í , EuÎH, ïuï(x) £ Eu Ù Ex, for all xÎDu}

then the H-valued model is

V^H =

If uÎV^H then u is called a set in V^H. We identifiy ïuï with u.

Let j(a₁, a₂, a₃, ... ,a_n) be a formula and j be a sentence. A formula of V^H is a formula of intuitionistic set theory with constants from elements of V^H.

Definition2: Let u, vÎV^H, j and y be sentences of V^H and j(a) be a formula of V^H. Then

i- úçEuúç = Eu

ii- úçuÎvúç = )

iii- úçu = vúç = ) Ù) Ù Eu Ù Ev

iv- úçj Ù yúç = úçjúçÙúçyúç

v- úçj Ú yúç = úçjúçÚúçyúç

vi- úçj ® yúç = úçjúç®úçyúç

vii- úçØjúç = Øúçjúç

viii- úç"j(x) úç = úçj(x) úç)

ix- úç$j(x) úç = Ùúçj(x) úç)

We abbreviate "x(xÎu®j(x)) to ("xÎu)j(x) and $x(xÎuÙj(x)) to ($xÎu)j(x) resp.

Lemma 1: Let j(a) be a formula of V^H and uÎ V^H.

i- úç"xj(x) úç = úçj(x) úç) = úçj(x) úç)

ii- úç$xj(x) úç = úçj(x) úç) = úçj(x) úç)

Thus for u, vÎ V^H,

úçu Í vúç = úçxÎvúç) = úçxÎvúç)

Let u, vÎV^H. If úçuÍvúç=1 then we say u is a subset of v in V^H and write uôv. If uôv and v ô u then ve say u and are similar and denote it as u:v.

Lemma 2: Let u, v Î V^H

i- = Ù Ù Eu Ù Ev

ii- u ôv iff £ for all x Î V^H

iii- u ~ v iff = for all x Î V^H

Definition 3: For u, v Î V^H the cartesian product of u and v in V^H is defined in the following way,

D(uv) = {: x Î Du, y Î Dv}, E(uv) = Eu Ù Ev and (uv) : → u(x) Ù v(y)

We will write instead of .

From the above definition we write,

= Ù

Proposition 1: All axioms of intuitionistic set theory are valid in V^H.

From the above proposition, it is show that V^H is a model of intuitionistic set theory.

Definition 4: Let u, v Î V^H be sets. The intersection and union of u and v are defined as following.

i- D(uv) = DuDv, E(uv) = Eu Ù Ev and u v : x → Ù

ii- D(uv) = Du Dv, E(uv) = Eu Ú Ev and u v : x → Ú

Lemma 3: For all x, u, v Î V^H , the followings hold.

i- = Ù

ii- = Ú

Lemma 4: For all u, v, w Î V^H ,

i- uv ô u, uv ô v and if w ô u and w ô v then w ô uv

ii- u ô uv, u ô uv and if u ô w and v ô w then uv ô w

2. Relations in the model V^H

Definition 5: The relation in V^H is a subset of a cartesian product of two sets in V^H.

For R, u, v Î V^H , R is a relation from u to v in V^H iff R u´v. If u = v then R is called the relation on u. We often write xRy instead of ÎR.

Definition 6: Let u Î V^H be a set. The identity relation I_u on u defined by

D(I_u) = {: xÎDu}, E(I_u) = Eu and I_u: →u(x)

I_u is also called the equality relation on u.

Lemma 5: Let u Î V^H be a set.

i- = Ù

ii- = , for all zÎV^H

iii- I_u ô u ´ u, that is I_u is a relation on u.

Definition 7: For R,S ÎV^H, the composition SoR in V^H is defined by

D(SoR) = {: x,z Î V_b^H}, E(RoS) = ERÙES and SoR: →

where b Î O_n such that DR DS V_b^H

Lemma 6: Let R, S, u, v, w ÎV^H be sets. Then the following holds.

i- = , for all x,y Î V^H.

ii- =, for all t Î V^H.

iii- If R ô u ´ v and Sôv ´ w then SoR ô u ´ w.

iv- (ToS)oR = To(SoR).

v- If R ô u ´ v then RoI_u ~ R~I_voR.

Definition 8: For R Î V^H, the inverse relation R^-1 of R in V^H is definite by

D(R^-1) = {: x,y Î V^H}, E(R^-1) = ER and R^-1 : → .

where b is an ordinal which satisfies DR V_b^H.

Lemma 7: Let R, u, v, then the following holds.

i- .

ii- = , for all w Î V^H.

iii- If R ô u ´ v then R^-1 ô v ´ u.

iv- (u ´ v)^-1 ~ v ´ u.

Proposition 2: Let R,S, u,v Î V^H be sets then the following conditions are hold.

i- ()^-1 ô R, if R ô u´v then ()^-1 ~ R.

ii- ~ Ç.

iii- ~ o.

iv- If R ô S then ô .

The most of the properties of relations on V^H are defined same way as in V.

Definition 9: u Î V^H and R be a relation on u in V^H.

i- R is reflexive iff úç ("xÎu)(xRx) úç=1.

ii- R is symmetric iff úç ("x"y)(xRy ® yRx) úç=1.

iii- R is transitive iff úç ("x"y"z)(xRyÙyRz ® xRz) úç=1.

iv- R is connected iff úç ("x"y)(xRy Ú yRx) úç=1.

The relation on a set is equivalence relation in V^H iff it is reflexive, symmetric and transitive. For aÎu, the set is defined by

D() = {y:yÎ}, E=Ea, and :x®R(a,x)

Thus, we define the set U/R with,

D(u/R)={ :aÎu}, E(u/R) = EuÙER, and u/R: ®

It is called the quotient set of u by R.

Lemma 8: Let R be a relation on uÎ V^H

i- R is reflexive iff I_u ô R.

ii- R is symmetric iff ô R iff R ~ .

iii- R is connected iff u´v ô RÈ iff u´u : RÈ.

Definition 10: Let R be a relation from u to v in V^H

i- R is total if úç ("xÎu)($yÎv)(xRy) úç=1.

ii- R is surjective if úç ("yÎv)($xÎu)(xRy) úç=1.

iii- R is injective if úç ("x,yÎu)("zÎv)(xRz Ù yRz ® x = y) úç=1.

iv- R is univalent if úç ("xÎu)("y,zÎv)(xRy Ù xRz ® y = z) úç=1.

For each uÎV^H, the identity relation I_u on u is defined by:

D(I_u)={<xx>: xÎDu}, E(I_u)=Eu and I_u : <xx>®u(x)

Proposition 3: Let R be a relation from u to v in V^H.

i- R is total iff I_u ô oR.

ii- R is surjective iff I_v ô Ro.

iii- R is injective iff oRôI_u.

iv- R is univalent iff RoôI_v.

Proof: Let R be a relation from u to v in V^H.

i- Since = úç ("xÎu)x(oR)x úç=úç ("xÎu)($yÎv) (xRy) úç, R is total iff =1 iff I_u ô oR.

ii- In V^H, oRÍI_u iff ("x, yÎu)(x(oR)y®x=y) iff ("x, yÎu)($zÎv)(xRzÙyRz)®x=y) iff R is injective.

iii and iv similar to i and iii respectively.

It is obvious that the inverse of a total (resp. surjective) relation is surjective (resp. total) and the inverse of an injective (resp. univalent) relation is univalent (resp. injective). A composition of total (resp. surjective, injectivei univalent) relations is total (resp. surjective, injective, univalent)

3. Mappings in model V^H

Definition 11: Let u, v, fÎ V^H, f is called a mapping from u to v in V^H if f is total and univalent relation from u to v in V^H.

We write f:u®v in V^H if f is a mapping from u tov in V^H. Hence f:u®v in V^H iff úçf:u®vúç=1, that is,

úçf Í u´vúç=úç ("x$y)(<xy>Îfúç = úç ("x"y"z)(<xy>ÎfÙ<xz>Îf)®(y=z) úç=1.

We often write f(x) = y instead of <xy> Î f or xfy. By proposition1.2.(1)(4), a relation f from u to v is a mapping iff I_u ô and ô I_v .

A composition of two mappings is also a mappings, that is, if f is a mapping from u to v and g is a mapping from v to w, then the composition gof is a mapping from u to w in V^H. Identity relations are always mappings.

For a mapping f from u to v in V^H, the composition R_f = is always an equivalence relation. Then we have úçxR_fyúç = úçf(x)=f(y) úç for all x, yÎV, where, f(x) = f(y) is an abbreviation for $z(xfz Ù yfz).

An injection (resp. Surjection) is an injective (resp. Surjective) mapping, and a bijection is an injective and surjective mapping. By proposition 3 we have

Lemma 9: Let f be a relation from u to v in V^H

i- f is an injection iff I_u : and ô I_v.

ii- f is a surjection iff I_u ô and : I_v.

iii- f is bijection iff I_u : and : I_v.

Hence for a mapping f:u®v in V^H, f is injective iff I_u : and is surjective iff : I_v.

Theorem 1: Let u,vÎ V^H be sets, f:u®v is a function in V^H then there is a decomposition of f such that

f ~ kohog

where k:f(u)®v is an injection, h:®f(u) is a bijection and g:u® is a surjection.

Proof: Let úç<a>Îgúç=úç<ab>ÎI_uúç=úça=búç then

úçÍúç=úç (",Î)(<>ή=úç

= úç ("$d)(<d>Îg Ù <d>Îg ®=úç

= úç ("$d)( úçd=búç Ù úçd=cúç ®=úç=1

therefore ô.

úçg Í u´úç= úç ("a$)(<a> Îgúç= úç ("a$)(<a>ÎgÙ<a>Îg®=úç=1.

From the same way, if we define the function h: ®f(u) with

úç<y>Îhúç=

we obtained h is a bijection.

If we choose k is a embedding mapping then it is clear that k is an injection.

We proved only f ô kohog,

úçfÍ kohogúç=(f(x,y)®(x,y)Î kohog)

= (f(x,y)®($zÎf(u), k(z,y)Ù(x,z)Î hog)

= (f(x,y)® ($zÎf(u), Î k(z,y)Ù(x, )Îg(,z)Î h)

= (f(x,y)® ($zÎf(u), Î,(z=y)Ù(x=a)Ù((m=a)Ù(m,z)Îf))

³ (f(x,y)® (($z,),(z=y)Ù(x=a)Ù(a,z)Îf)

= (f(x,y)®f(x,y)) = 1

therefore f ô kohog.

The function g is called natural mapping which is defined above theorem and write natR_f instead of g.

Theorem 2: Let u,vÎV^H be sets. R be an equivalence relation on u, and f:u®v is a function in V^H. If RôR_f then there exist a function :®v in V^H such that o natR_f :f.

Proof: We define úç<,y>Îúç=úç<by>Îfúç. From this

úç<x,y>ÎonatRúç= úç$, <x, >ÎnatRÙ<y>Îúç

=(úçÎA/RúçÙúç<x, >ÎnatRúçÙúç<y>Îúç)

£ (úçÎA/RúçÙúçx=aúçÙúç<ay>Îfúç)

£ (úçaÎAúçÙúçx=aúçÙúç<ay>Îfúç)

£ úç<xy>Îfúç.

Therefore, f ô onatR.

The converse is clear from the definition of

úçfÍonatRúç=(<xy>Îf®<xy>ÎonatR).

References

[1] Atanassov, Intuitionistic fuzzy sets, Phiysica-Verlag, Heidelberg, New York, 1999.

[2] Scott, Identity and existence in intuitionistic logic. In: MP Fourman, et. al., Application of Sheaves., 753 (1979), Lecture Notes in Mathematics, 660-696

[3] Shimoda, M., A natural interpretation of fuzzy sets and fuzzy relations, (128) (2002), Fuzzy Sets and Systems, 135-147.

[4] Titani, S., Completeness of global intuitionistic set theory, 62 (2) (1997) J. Symbolic Logic, 506-528.

[5] Zadeh, L.A., Fuzzy sets, Inform and Control 8 (3) (1965) 338-353.

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