ON INTERRELATIONS BETWEEN FUZZY ...

New Mathematics and Natural Computation Vol. 8, No. 3 (2012) 297310 # .c World Scienti¯c Publishing Company DOI: 10.1142/S1793005712500044

ON INTERRELATIONS BETWEEN FUZZY CONGRUENCE AXIOMS

S. R. CHAUDHARI* and S. S. DESAI† Department of Mathematics, Shivaji University Vidyanagar, Kolhapur (M.S.)-416004, India *[email protected] † santosh_desai21@rediﬀmail.com

In this paper we establish interrelations between fuzzy direct revelation axiom (FDRA), fuzzy transitive-closure coherence axiom (FTCCA), fuzzy consistent-closure coherence axiom (FCCCA) and fuzzy intermediate congruence axiom (FICA). We also establish their relationships with weak fuzzy congruence axiom (WFCA), strong fuzzy congruence axiom (SFCA) and weak axiom of fuzzy revealed preference (WAFRP). Condition for equivalence of fuzzy Arrow axiom (FAA) and weak fuzzy congruence axiom (WFCA) on arbitrary domain is also given. Keywords: Fuzzy preference relation; fuzzy choice function; fuzzy consistent closure; congruence axioms; fuzzy Arrow axiom.

1. Introduction To study the rational behavior of consumer, Samuelson20 introduced the concept of revealed preference theory. Uzawa26 and Arrow1 have abstractly studied Samuelson's theory of revealed preference by introducing the concept of choice function. The work of Uzawa and Arrow was continued by Sen2123 and many others14,19,24,25 by considering the domain of the choice function consisting of all ¯nite non-empty subsets of the set of alternatives. They studied rationality of choice function with the help of properties of revealed preference axiom, the consistency conditions and the congruence axioms. Orlovsky18 studied fuzzy preference relation. The work of Orlovsky was continued by De Baets et al.,9 Barrett et al.46 and Fodor et al.10 Banerjee2 fuzzi¯es only range of choice functions and introduced three independent fuzzy congruence conditions to characterize rationality of fuzzy choice function. Wang27 establishes connections between congruence conditions introduced by Banerjee and shows they are not independent. Georgescu11,12 and Mordeson et al.17 have studied fuzzy choice theory. Mordeson et al.17 have studied fuzzy choice functions in di®erent angle. In their work, they have shown fuzzy choice functions satisfying and are rationalizable and also they satisfy Fuzzy Arrow Axiom (FAA) and Weak Axiom of Revealed Preference

297

298

S. R. Chaudhari & S. S. Desai

(WARP). In Ref. 11, Georgescu studied fuzzy choice functions in a more general form. The domain and co-domain of a Georgescu choice functions are made of fuzzy sets of alternatives. Georgescu11 formulates fuzzy forms of revealed preference axioms, consistency conditions and the congruence axioms, etc. and studied rationality of fuzzy choice functions. He has extended results of Ritcher19 and Suzumura24,25 by introducing notions like M-rational, G-rational, M-normal, G-normal and full rationality of fuzzy choice function. In Ref. 12, Georgescu introduced indicator of fuzzy Arrow axiom and proved equivalence between full rationality and fuzzy Arrow axiom in a more generalized form. The whole work of Georgescu is based on the following two hypotheses: (H1) For every S 2 B, CðSÞ are normal fuzzy subset of X. (H2) B includes all the fuzzy sets ½x1 ; x2 ; . . . ; xn ; n 1 and x1 ; x2 ; . . . ; xn 2 X, where B is domain of the fuzzy choice function and ½x1 ; x2 ; . . . ; xn is characteristic function of fx1 ; x2 ; . . . ; xn g. The concept of rational choice as an optimizing choice is binary in nature in that the choice from any set is to be accounted for in terms of binary relation. Therefore in Refs. 7 and 8, we have studied rationality of fuzzy choice function on base domaina by de¯ning fuzzy direct revelation axiom (FDRA), fuzzy transitive-closure coherence axiom (FTCCA), fuzzy consistent-closure coherence axiom (FCCCA) and fuzzy intermediate congruence axiom (FICA). However, throughout this paper we have waved the general domain as well as the base domain to consider more appropriate arbitrary domain for fuzzy choice function. In Sec. 2, we recall de¯nitions and properties of residuation, fuzzy preference relation, fuzzy consistent closure and fuzzy choice function. In Sec. 3, we established interrelation between fuzzy direct revelation axiom (FDRA), fuzzy transitive-closure coherence axiom (FTCCA), fuzzy consistent-closure coherence axiom (FCCCA) and fuzzy intermediate congruence axiom (FICA). Also, interrelationship between FDRA, FTCCA, FCCCA, FICA and WFCA, SFCA, WAFRP are established. Georgescu11 shows that the fuzzy Arrow axiom (FAA) and weak fuzzy congruence axiom (WFCA) are equivalent on the domain B ¼ F ðXÞnfg under the hypotheses H1 and H2. In this section, we establish a su±cient condition so that they are equivalent on arbitrary domain. Section 4 mainly devoted to examples shows that some inequalities of the previous section do not hold. 2. Fuzzy Preference Relations and Choice Functions In this section, we recall those concepts of fuzzy preference relations and fuzzy choice functions which are required in the subsequent part of this paper.

a Base Domain: A domain of a fuzzy choice function, B, is said to be base domain if it contains all fuzzy sets

of the type ½xi ; ½xj ; xk , i; j; k ¼ 1; 2; . . . ; n for x1 ; x2 ; . . . ; xn 2 X.

On Interrelations Between Fuzzy Congruence Axioms

299

Recall that for any a; ai ; b; bi 2 ½0; 1, we denote a _ b ¼ maxða; bÞ, a ^ b ¼ minða; bÞ, V V i2I ai ¼ inffai : i 2 Ig. Clearly, i2I ai ai , 8 i 2 I

_i2I ai ¼ supfai : i 2 Ig and and _i2I ai ai , 8 i 2 I.

A triangular norm (or t-norm) is a binary relation on [0, 1] such that (i) a b ¼ b a (ii) a ðb cÞ ¼ ða bÞ c, (iii) a b a c; when b c and (iv) a 1 ¼ a, for all a; b; c 2 ½0; 1: If satis¯es (i)(iii) and a 0 ¼ a, for all a; b; c 2 ½0; 1 then it is called triangular co-norm (or s-norm). We say that a t-norm is continuous, if it is continuous as a function on [0, 1]. For any continuous t-norm , a binary operation ! on [0, 1] de¯ned as a ! b ¼ supfc 2 ½0; 1 : a c bg is called the residuum or the fuzzy implication associated with . The biresiduum operation $ on [0, 1] de¯ned by a $ b ¼ ða ! bÞ ðb ! aÞ: If a b ¼ maxð0; a þ b 1Þ, then is t-norm (called Lukawiewicz t-norm) and a ! b ¼ minð1; 1 a þ bÞ is the associated fuzzy implication, known as Lukawiewicz fuzzy implication. Similarly if a b ¼ minða; bÞ; then is G€odel t-norm and 1 if a b a!b¼ b if a > b is the associated fuzzy implication. For other t-norms and their associated fuzzy implications, we refer to Refs. 3, 15 and 16.

Lemma 2.1.11,13,15,16 If ! is a fuzzy implication associated with a continuous t-norm on [0, 1], then (i) (ii) (iii) (iv) (v) (vi) (vii)

a ða ! bÞ a ^ b abc,ab!c ab,a!b¼1 1!a¼a a!1¼1 a b ) b ! c a ! c and c ! a c ! b; ða ! bÞ ðb ! cÞ a ! c

If is a continuous t-norm, then a unary operation : on [0, 1] de¯ned as :a ¼ a ! 0 ¼ supfc 2 ½0; 1 : a c ¼ 0g is called negation associated with t-norm on [0, 1]. Note that 1 if a ¼ 0 :a ¼ 1 a and :a ¼ 0 if a > 0 are negations associated with Lukawiewicz and G€odel t-norm respectively. Lemma 2.2.11,13,15 If : is a negation associated with a continuous t-norm , then for any a; b 2 ½0; 1 (i) a :b , a b ¼ 0; (ii) a :a ¼ 0

300


(iii) a ! b :b ! :a (iv) a b ) :b :a (v) :ða _ bÞ ¼ :a ^ :b; :ða ^ bÞ ¼ :a _ :b. Lemma 2.3.11,13,15,16 For fai gi2I ½0; 1 and a 2 ½0; 1 we have (i) ð_i2I ai Þ a ¼ _i2I ðai aÞ V V V (ii) ð i2I ai Þ ð j2I bj Þ i;j2I ðai bj Þ Let X be a non-empty set. Then A : X ! ½0; 1 is called a fuzzy set of X and F ðXÞ is the set of all fuzzy sets on X. A fuzzy set A is called normal, if AðxÞ ¼ 1, for some x 2 X and is called empty, if AðxÞ ¼ 0, for all x 2 X. We denote empty fuzzy set by . For any x1 ; x2 ; . . . ; xn 2 X, we shall denote characteristic function of fx1 ; x2 ; . . . ; xn g by ½x1 ; x2 ; . . . ; xn . Thus 1 if y 2 fx1 ; x2 ; . . . ; xn g ½x1 ; x2 ; . . . ; xn ðyÞ ¼ : 0 otherwise V For A; B 2 F ðXÞ let us denote IðA; BÞ ¼ z2X ½AðzÞ ! BðzÞ and EðA; BÞ ¼ V z2X ½AðzÞ $ BðzÞ. IðA; BÞ is called subset hood degree of A in B and EðA; BÞ is called degree of equality of A and B. A fuzzy set Q of X Y is called fuzzy relation from X to Y . If X ¼ Y , then Q is called fuzzy binary relation on X and Qðx; yÞ denotes the degree of preference of x in relation to y. Therefore fuzzy binary relation on X is some time called fuzzy preference relation on X. We denote strict fuzzy preference relation by P ðQÞ and de¯ne it by P ðQÞðx; yÞ ¼ Qðx; yÞ :Qðy; xÞ; for all x; y 2 X. If Qðx; yÞ Qðy; zÞ Qðx; zÞ, for all x; y; z 2 X, then Q is called -transitive and Q is the transitive closure of Q and is given by ( ) W W Qðx; yÞ ¼ Qðx; yÞ _ ½Qðx; z1 Þ Qðz1 ; z2 Þ Qðzk ; yÞ : k2N z1 ; z2 ;...;zk 2X

It is clear for the fuzzy preference relation Q the following statements are equivalent. (i) Q is transitive, (ii) Q Q Q, (iii) Q ¼ Q. Lemma 2.4. The relation Q possesses the following properties (i) Q is the smallest transitive relation containing Q (ii) Q1 Q2 ) Q1 Q2 (iii) Q ¼ Q ^ of a fuzzy Now we recall, from Ref. 7, the concept of fuzzy consistent closure Q preference relation Q, which we use to de¯ne congruence axioms in the following Sec. 3. De¯nition 2.5.7 Let Q be a fuzzy preference relation on X. The fuzzy consistent ^ and is given by closure of Q is denoted by Q ^ yÞ ¼ Qðx; yÞ _ ½Qðx; yÞ Qðy; xÞ; Qðx;

for all x; y 2 X


301

The following are the basic properties of fuzzy consistent closure. Lemma 2.6.7 Let Q be any fuzzy preference relation, then (i) (ii) (iii)

^ Q Q ^1 Q ^2 Q1 Q2 ) Q ^ QQ

Remark 2.7. If the fuzzy preference relation Q is transitive, then the fuzzy ^ and fuzzy preference relation Q coincides. consistent closure Q De¯nition 2.8.11 A fuzzy choice space is a pair ðX; BÞ, where X is a non-empty set called set of alternatives and B is a non-empty family of non-zero fuzzy subsets of X called set of available alternatives. A fuzzy choice function (or fuzzy consumer) on ðX; BÞ is a function C : B ! F ðXÞ such that for each S 2 B, CðSÞ is non-zero and CðSÞ S. In the language of fuzzy consumer, X is called set of bundles, B is called family of fuzzy budgets, the pair ðX; BÞ is called fuzzy budget space. CðSÞðxÞ denotes the degree to which the bundle x is chosen subject to the fuzzy budget S. In classical choice theory, it is always assumed that every, available set S must have at least one choice i.e. CðSÞ is non-empty, for every S, to meet this requirement, throughout this paper, we assume that the fuzzy choice function C must be normal, i.e. for every S 2 B, CðSÞðxÞ ¼ 1, for some x 2 X. Thus, in our approach the domain of fuzzy choice function is arbitrary with the hypothesis H1 (i.e. every S 2 B and CðSÞ are normal fuzzy subsets of X) as given by Georgescu in Ref. 11. De¯nition 2.9.11 Let C : B ! F ðXÞ be a fuzzy choice function on ðX; BÞ. Then we de¯ne four fuzzy revealed preference relations on X as follows W Rðx; yÞ ¼ ½CðSÞðxÞ SðyÞ; P ðx; yÞ ¼ Rðx; yÞ :Rðy; xÞ; S2B

Iðx; yÞ ¼ Rðx; yÞ Rðy; xÞ

and

P~ ðx; yÞ ¼

W

½CðSÞðxÞ SðyÞ :CðSÞðyÞ:

S2B

We shall call R the fuzzy revealed preference relation generated by C, P the strict fuzzy revealed preference relation generated by C, I the indi®erent fuzzy revealed preference relation generated by C and P~ the strong fuzzy revealed preference relation generated by C. Now, we shall recall weak fuzzy congruence axiom (WFCA), strong fuzzy congruence axiom (SFCA), fuzzy Arrow axiom (FAA) and weak axiom of fuzzy revealed preference (WAFRP) discussed in Ref. 11. Weak Fuzzy Congruence Axiom ðWFCAÞ: For any S 2 B and x; y 2 X Rðx; yÞ CðSÞðyÞ SðxÞ CðSÞðxÞ Strong Fuzzy Congruence Axiom ðSFCAÞ: For any S 2 B and x; y 2 X Rðx; yÞ CðSÞðyÞ SðxÞ CðSÞðxÞ

302


Fuzzy Arrow Axiom ðFAAÞ: For any S; T 2 B and x 2 X IðS; T Þ SðxÞ CðT ÞðxÞ EðS \ CðT Þ; CðSÞÞ Weak Axiom of Fuzzy Revealed Preference ðWAFRPÞ: For any x; y 2 X P~ ðx; yÞ :Rðy; xÞ Lemma 2.10.11 Let C : B ! F ðXÞ be a fuzzy choice function on ðX; BÞ. Then (a) WFCA implies WAFRP (b) WFCA and FAA are equivalent (c) SFCA implies WFCA 3. Interrelations In Refs. 7 and 8, we have studied rationality of fuzzy choice function by introducing various fuzzy axioms namely fuzzy direct revelation axiom (FDRA), fuzzy transitive closure coherence axiom (FTCCA), fuzzy consistent closure coherence axiom (FCCCA) and fuzzy intermediate congruence axiom (FICA). We have obtained few results which connect G-rationality, full rationality, G-normality and above fuzzy axioms. We believe that the further study of fuzzy choice functions will be in°uenced, if one discusses the interrelation between the fuzzy axioms introduced so far. This is precisely the aim of this section. We begin with the fuzzy axioms introduced in Refs. 7 and 8. Fuzzy Direct Revelation Axiom ðFDRAÞ: For any S 2 B and x 2 X V SðxÞ ½SðzÞ ! Rðx; zÞ CðSÞðxÞ z2X

Fuzzy Transitive-closure coherence Axiom ðFTCCAÞ: For any S 2 B and x 2 X V SðxÞ ½SðzÞ ! Rðx; zÞ CðSÞðxÞ z2X

Fuzzy Consistent-closure coherence Axiom ðFCCCAÞ: For any S 2 B and x 2 X V ^ zÞ CðSÞðxÞ SðxÞ ½SðzÞ ! Rðx; z2X

Fuzzy Intermediate Congruence Axiom ðFICAÞ: For any S 2 B and x; y2X ^ yÞ CðSÞðyÞ SðxÞ CðSÞðxÞ Rðx; Remark 3.1. If the fuzzy revealed preference relation R is transitive, then R; R and ^ coincide and above axioms reduces to a fuzzy direct revelation axiom. R In the following two theorems we establish interrelations between these four fuzzy axioms and their relations with WFCA, SFCA and WAFRP. Theorem 3.2. Let C : B ! F ðXÞ be a fuzzy choice function with an arbitrary nonempty domain B. Then (a) FTCCA implies FICA


303

(b) FICA implies FCCCA (c) FCCCA implies FDRA Proof. (a) Let S 2 B and x; y 2 X. Then by Lemmas 2.6(iii) and 2.4(i), one gets ^ yÞ CðSÞðyÞ SðzÞ Rðx; ^ yÞ Rðy; zÞ Rðx; yÞ Rðy; zÞ Rðx; zÞ: Rðx; By Lemma 2.1(ii), we have ^ yÞ CðSÞðyÞ SðzÞ ! Rðx; zÞ; for any z 2 X: Rðx; V ^ yÞ CðSÞðyÞ SðxÞ SðxÞ z2X ½SðzÞ ! Rðx; zÞ CðSÞðxÞ: T h e r e f o r e , Rðx; ^ yÞ CðSÞðyÞ SðxÞ CðSÞðxÞ. Thus, Rðx; (b) Let S 2 B and x; y 2 X. Since the fuzzy choice function C is normal, we have for S 2 B there exists t 2 X such that CðSÞðtÞ ¼ 1: Therefore SðtÞ ¼ 1. Thus, by Lemma 2.1(iv), we have Therefore, V ^ zÞ SðxÞ ½SðtÞ ! Rðx; ^ tÞ SðxÞ ½SðzÞ ! Rðx; z2X

^ tÞ CðSÞðtÞ SðxÞ ¼ Rðx; CðSÞðxÞ:

Therefore, FICA implies FCCCA. ^ Therefore, by Lemma 2.1(vi) for any (c) By Lemma 2.6(i), we have R R. S 2 B and x 2 X we have ^ yÞ SðyÞ ! Rðx; yÞ SðyÞ ! Rðx;

for all y 2 X:

Therefore, SðxÞ

V

½SðzÞ ! Rðx; zÞ SðxÞ

z2X

V

^ zÞ ½SðzÞ ! Rðx;

z2X

CðSÞðxÞ: Hence, FCCCA implies FDRA. Theorem 3.3. Let C : B ! F ðXÞ be a fuzzy choice function with an arbitrary nonempty domain B. Then (a) (b) (c) (d) (e)

FTCCA and SFCA are equivalent FICA implies WFCA WFCA implies FDRA FICA implies WAFRP SFCA implies FICA

Proof. (a) Let S 2 B and x; y 2 X: Then for any z 2 X, we have Rðx; yÞ CðSÞðyÞ SðzÞ Rðx; yÞ Rðy; zÞ Rðx; yÞ Rðy; zÞ Rðx; zÞ: Thus, Rðx; yÞ CðSÞðyÞ SðzÞ ! Rðx; zÞ, for all z 2 X, by Lemma 2.1-(ii).

304


V Therefore, Rðx; yÞ CðSÞðyÞ SðxÞ SðxÞ z2X ½SðzÞ ! Rðx; zÞ CðSÞðxÞ: This proves FTCCA implies SFCA. Now, for any S 2 B. We have by normality, CðSÞðtÞ ¼ 1 and hence SðtÞ ¼ 1; for some t 2 X. V Thus, z2X ½SðzÞ ! Rðx; zÞ SðtÞ ! Rðx; tÞ ¼ Rðx; tÞ, by Lemma 2.1(iv). V Therefore, SðxÞ z2X ½SðzÞ ! Rðx; zÞ Rðx; tÞ CðSÞðtÞ SðxÞ CðSÞðxÞ: Hence SFCA implies FTCCA (b) By Lemma 2.6(i) and FICA. (c) Let S 2 B. Then by normality of C, we have CðSÞðtÞ ¼ 1 and hence SðtÞ ¼ 1, for some t 2 X: V Thus, z2X ½SðzÞ ! Rðx; zÞ SðtÞ ! Rðx; tÞ ¼ Rðx; tÞ by Lemma 2.1(iv). V Then, SðxÞ z2X ½SðzÞ ! Rðx; zÞ Rðx; tÞ CðSÞðtÞ SðxÞ CðSÞðxÞ: V Therefore, SðxÞ z2X ½SðzÞ ! Rðx; zÞ CðSÞðxÞ: (d) Let x; y 2 X. Then by Lemma 2.6(i), we have W ^ xÞ ^ xÞ ¼ ½CðSÞðxÞ SðyÞ :CðSÞðyÞ Rðy; P~ ðx; yÞ Rðy; xÞ P~ ðx; yÞ Rðy; S2B W ^ xÞ CðSÞðxÞ SðyÞ :CðSÞðyÞ: ½Rðy; ¼ S2B

^ yÞ CðSÞðxÞ SðyÞ CðSÞðyÞ, for all S 2 B. By FICA, we have Rðx; ~ Thus, P ðx; yÞ Rðy; xÞ _S2B ½CðSÞðyÞ :CðSÞðyÞ ¼ 0: Therefore, by Lemma 2.2(i), P~ ðx; yÞ :Rðy; xÞ: (e) By Lemma 2.6(iii) and SFCA. Georgescu11 shows that fuzzy Arrow axiom and weak fuzzy congruence axiom are equivalent on domain B ¼ F ðXÞnfg under the hypotheses H1 and H2. In the following example ¯rst we show FAA need not imply WFCA on arbitrary domain. We note that WFCA implies FAA on any domain. Example 3.4. Let X ¼ fa; b; c; dg. De¯ne a fuzzy choice function C on B ¼ f½a; b; c; ½a; b; d; ½a; c; b; ½b; dg as follows Cð½a; b; cÞðaÞ ¼ 0:8; Cð½a; b; cÞðbÞ ¼ 1; Cð½a; b; cÞðcÞ ¼ 1; Cð½a; b; dÞðaÞ ¼ 1; Cð½a; b; dÞðbÞ ¼ 0:7; Cð½a; b; dÞðdÞ ¼ 0:6; Cð½a; c; dÞðaÞ ¼ 1; Cð½a; c; dÞðcÞ ¼ 1; Cð½a; c; dÞðdÞ ¼ 1; Cð½b; dÞðbÞ ¼ 0:9; Cð½b; dÞðdÞ ¼ 1: Then the fuzzy revealed preference relation R is given by

Here C satis¯es fuzzy Arrow axiom.


305

But Rða; bÞ ^ Cð½a; b; cÞðbÞ ^ ½a; b; cðaÞ ¼ 1 and Cð½a; b; cÞðaÞ ¼ 0:8. This shows that FAA does not imply WFCA on arbitrary domain. Following theorem shows that if the domain of fuzzy choice function is closed under intersection, then FAA and WFCA are equivalent on arbitrary domain also. Theorem 3.5. Let C : B ! F ðXÞnfg be a fuzzy choice function with an arbitrary non-empty domain B which is closed under intersection. Then fuzzy Arrow axiom and weak fuzzy congruence axiom are equivalent for G€ odel t-norm ^: Proof. To prove FAA, it is su±cient to prove IðS; T Þ ^ SðxÞ ^ CðT ÞðxÞ CðSÞðzÞ $ SðzÞ ^ CðT ÞðzÞ, for any z 2 X and S; T 2 B. Let S; T 2 B and x; y 2 X. Since IðS; T Þ SðzÞ ! T ðzÞ for all z 2 X, by Lemma 2.1(ii), we have IðS; T Þ ^ SðzÞ T ðzÞ. Thus, IðS; T Þ ^ SðxÞ ^ CðT ÞðxÞ ^ CðSÞðzÞ ¼ IðS; T Þ ^ SðzÞ ^ SðxÞ ^ CðT ÞðxÞ ^ CðSÞðzÞ ^ SðzÞ T ðzÞ ^ SðxÞ ^ CðT ÞðxÞ ^ CðSÞðzÞ ^ SðzÞ ¼ CðSÞðzÞ ^ SðxÞ ^ CðT ÞðxÞ ^ T ðzÞ ^ SðzÞ Rðz; xÞ ^ CðT ÞðxÞ ^ T ðzÞ ^ SðzÞ CðT ÞðzÞ ^ SðzÞ; by WFCA then by Lemma 2.1(ii) IðS; T Þ ^ SðxÞ ^ CðT ÞðxÞ CðSÞðzÞ ! CðT ÞðzÞ ^ SðzÞ:

ð3:1Þ

By the normality of fuzzy choice function we have CðSÞðtÞ ¼ 1 and SðtÞ ¼ 1 for some t 2 X: Thus, IðS; T Þ ^ SðxÞ ^ CðT ÞðxÞ ^ SðzÞ ^ CðT ÞðzÞ ¼ IðS; T Þ ^ SðtÞ ^ SðxÞ ^ CðT ÞðxÞ ^ SðzÞ ^ CðT ÞðzÞ ^ CðSÞðtÞ T ðtÞ ^ SðxÞ ^ CðT ÞðxÞ ^ SðzÞ ^ CðT ÞðzÞ ^ CðSÞðtÞ CðT ÞðzÞ ^ T ðtÞ ^ CðSÞðtÞ ^ SðzÞ Rðz; tÞ ^ CðSÞðtÞ ^ SðzÞ CðSÞðzÞ; by WFCA: Thus, IðS; T Þ ^ SðxÞ ^ CðT ÞðxÞ ^ SðzÞ ^ CðT ÞðzÞ CðSÞðzÞ: Again by using Lemma 2.1(ii) one has IðS; T Þ ^ SðxÞ ^ CðT ÞðxÞ SðzÞ ^ CðT ÞðzÞ ! CðSÞðzÞ: Therefore, by (3.1) and (3.2) we have IðS; T Þ ^ SðxÞ ^ CðT ÞðxÞ CðSÞðzÞ $ SðzÞ ^ CðT ÞðzÞ This proves WFCA implies FAA.

for any z 2 X:

ð3:2Þ

306


Let S 2 B and x; y 2 X: Then Rðx; yÞ ^ CðSÞðyÞ ^ SðxÞ ¼ ¼

W

½CðT ÞðxÞ ^ T ðyÞ ^ CðSÞðyÞ ^ SðxÞ

T 2B

W

½CðT ÞðxÞ ^ T ðyÞ ^ CðSÞðyÞ ^ SðxÞ:

T 2B

ð3:3Þ

Since B is closed under intersection, for S; T 2 B, we have S \ T 2 B: Also S \ T S and S \ T T imply that IðS \ T ; SÞ ¼ 1 and IðS \ T ; T Þ ¼ 1. Thus, CðT ÞðxÞ ^ T ðyÞ ^ CðSÞðyÞ ^ SðxÞ ¼ ðIðS \ T ; SÞ ^ SðyÞ ^ T ðyÞ ^ CðSÞðyÞÞ ^ ðIðS \ T ; T Þ ^ SðxÞ ^ T ðxÞ ^ CðT ÞðxÞÞ EðCðS \ T Þ; S \ T \ CðSÞÞ ^ EðCðS \ T Þ; S \ T \ CðT ÞÞ ¼ EðCðS \ T Þ; T \ CðSÞÞ ^ EðCðS \ T Þ; S \ CðT ÞÞ V ½ðCðS \ T ÞðzÞ $ T ðzÞ ^ CðSÞðzÞÞ ^ ðCðS \ T ÞðzÞ $ SðzÞ ^ CðT ÞðzÞÞ z2X V ½ðSðzÞ ^ CðT ÞðzÞ ! CðS \ T ÞðzÞÞ ^ ðCðS \ T ÞðzÞ ! T ðzÞ ^ CðSÞðzÞÞ z2X

V

½SðzÞ ^ CðT ÞðzÞ ! T ðzÞ ^ CðSÞðzÞ; by Lemma 2:1ðviiÞ

z2X

SðxÞ ^ CðT ÞðxÞ ! T ðxÞ ^ CðSÞðxÞ: Therefore, by Lemma 2.1(ii) we get CðT ÞðxÞ ^ T ðyÞ ^ CðSÞðyÞ ^ SðxÞ ^ SðxÞ ^ CðT ÞðxÞ T ðxÞ ^ CðSÞðxÞ i.e. CðT ÞðxÞ ^ T ðyÞ ^ CðSÞðyÞ ^ SðxÞ T ðxÞ ^ CðSÞðxÞ CðSÞðxÞ;

for any T 2 B:

Therefore, by (3.3) we have Rðx; yÞ ^ CðSÞðyÞ ^ SðxÞ CðSÞðxÞ:

4. Examples In this section, we give examples which show (i) FCCCA does not imply FAA, (ii) WFCA does not imply FCCCA, (ii) FAA does not imply FDRA and (iv) FDRA does not imply FCCCA. Throughout this section we assume that ¼ ^ Example 4.1. Let X ¼ fa; b; c; dg. De¯ne a fuzzy choice function C on B ¼ f½a; b; d; ½b; ½b; c; Xg as follows Cð½a; b; dÞðaÞ ¼ 1; Cð½a; b; dÞðbÞ ¼ 0:8; Cð½a; b; dÞðdÞ ¼ 1; Cð½bÞ ðbÞ ¼ 1; Cð½b; cÞðbÞ ¼ 1; Cð½b; cÞðcÞ ¼ 0:9; CðXÞðaÞ ¼ 0:8; CðXÞðbÞ ¼ 0:8; CðXÞðcÞ ¼ 0:8 and CðXÞðdÞ ¼ 1:


307

Then the fuzzy revealed preference relation R is given by

Since R is not transitive, then its transitive closure R and fuzzy consistent closure ^ R are as follows

V ^ zÞ ¼ CðSÞðxÞ: Thus C For any S 2 B and x 2 X, SðxÞ ^ z2X ½SðzÞ ! Rðx; satis¯es FCCCA. But for ½a; b; d and X, Ið½a; b; d; XÞ ^ ½a; b; dðdÞ ^ CðXÞðdÞ ¼ 1 V and ½½a; b; dðzÞ ^ CðXÞðzÞ $ Cð½a; b; dÞðzÞ Cð½a; b; dÞðaÞ ! ½a; b; dðaÞ^ z2X CðXÞðaÞ ¼ 0:8. This shows that FCCCA does not imply FAA. Example 4.2. Let X ¼ fa; b; c; dg. De¯ne a fuzzy choice function C on B ¼ f½a; b; d; ½b; c; d; ½a; b; ½a; c; T g; where T ðaÞ ¼ 0; T ðbÞ ¼ 0:7; T ðcÞ ¼ 1; T ðdÞ ¼ 0 as follows Cð½a; b; dÞðaÞ ¼ 0:9; Cð½a; b; dÞðbÞ ¼ 1; Cð½a; b; dÞðdÞ ¼ 1; Cð½b; c; dÞðbÞ ¼ 1; Cð½b; c; dÞðcÞ ¼ 1; Cð½b; c; dÞðdÞ ¼ 1; Cð½a; bÞðaÞ ¼ 0:9; Cð½a; bÞðbÞ ¼ 1; Cð½a; cÞðaÞ ¼ 0:7; Cð½a; cÞðcÞ ¼ 1; CðT ÞðbÞ ¼ 0:7; CðT ÞðcÞ ¼ 1: From above fuzzy choice function C the fuzzy revealed preference relation R is given by

For any S 2 B and x; y 2 X; Rðx; yÞ ^ CðSÞðyÞ ^ SðxÞ CðSÞðxÞ i.e. C satis¯es ^ of R are given by, WFCA. The transitive closure R and fuzzy consistent closure R

308


V ^ zÞ ¼ 0:9 and CðSÞðaÞ ¼ 0:7. This For S ¼ ½a; c we get, SðaÞ ^ z2X ½SðzÞ ! Rða; shows that WFCA does not imply FCCCA. Remark 4.3. In Ref. 11, Georgescu shows that the fuzzy revealed preference relation R is transitive, if the fuzzy choice function C satis¯es WFCA, but the above example shows that on arbitrary domain R need not be transitive, when C satis¯es WFCA. Example 4.4. In Example 3.4, a fuzzy choice function C satis¯es FAA. But for V S ¼ ½a; b; d; SðbÞ ^ z2X ½SðzÞ ! Rðb; zÞ ¼ 0:9 and CðSÞðbÞ ¼ 0:7. This shows that FAA does not imply FDRA. Example 4.5. In Example 4.2, C satis¯es FDRA but for the choice set S ¼ ½a; c, V ^ zÞ ¼ 0:9 and CðSÞðaÞ ¼ 0:7. This shows that FDRA does SðaÞ ^ z2X ½SðzÞ ! Rða; not imply FCCCA.

5. Conclusion Rationality of fuzzy choice functions de¯ned on general domain was fully discussed in Ref. 11 along with interrelations between various fuzzy revealed and congruence axioms. Since the concept of rational choice as a optimizing choice is binary in nature, we have elsewhere7,8 discussed rationality of fuzzy choice function de¯ned on base domain along with the introduction of various fuzzy axioms namely fuzzy direct revelation axion (FDRA), fuzzy transitive-closure coherence axiom (FTCCA), fuzzy consistent closure coherence axiom (FCCCA) and fuzzy intermediate congruence axiom (FICCA). In this paper, we have discussed fuzzy choice functions de¯ned on more general domain, called arbitrary domain. This o®ers us to ¯nd interrelations between various fuzzy revelaed preference axiom and fuzzy congruence axiom introduced so far. To be speci¯c, we have established interrelationships shown in the following ¯gure.

The signi¯cance of these interrelations improves results discussed in Ref. 11 with the help of FDRA, FTCA, FCCCA and FICA.


309

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