An Intuitionistic fuzzy count and cardinality of Intuitionistic fuzzy sets

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Malaya Joural of Matematik 4(1)(2013) 123 133 A Ituitioistic fuzzy cout ad cardiality of Ituitioistic fuzzy sets B. K. Tripathy a, S. P. Jea b ad S. K. Ghosh c, a School of Computig Scieces ad Egieerig, V.

1 Malaya Joural of Matematik 4(1)(2013) A Ituitioistic fuzzy cout ad cardiality of Ituitioistic fuzzy sets B. K. Tripathy a, S. P. Jea b ad S. K. Ghosh c, a School of Computig Scieces ad Egieerig, V.I.T. Uiversity, Vellore , Tamiladu, Idia. b Departmet of Mathematics, Sailabala Wome s College, Cuttack, Odisha, Idia. c Departmet of Mathematics, Reveshaw Uiversity, Cuttack , Odisha, Idia. Abstract The otio of Ituitoistic fuzzy sets was itroduced by Ataassov 1 as a extesio of the cocept of fuzzy sets itroduced by Zadeh such that it is applicable to more real life situatios. I order to measure the cardiality of fuzzy sets several attempts have bee made 4,6,8. However, there are o such measures for ituitioistic fuzzy sets. I this paper we defie the sigma cout ad relative sigma cout for ituitioistic fuzzy sets ad establish their properties. Also, we illustrate the geratio of quatificatio rules. Keywords: Fuzzy set, Ituitioistic Fuzzy set, Ituitioistic fuzzy cout, Relative Ituitioistic fuzzy cout MSC: 57E05. c 2012 MJM. All rights reserved. 1 Itroductio The itroductio of the fuzzy cocept by Zadeh 7 is cosidered as a paradigm shift 5. It itroduces the cocept of graded membership of elemets istead of the biary membership used i Aristotelia logic. It is a very powerful modelig laguage that ca cope with a large fractio of ucertaities of real life situatios. Because of its geerality it ca be well adapted to differet circumstaces ad cotexts. The cardiality of a set i the crisp sese plays a importat role i Mathematics ad its applicatios. Similarly it is worthwhile to thik of cardiality of fuzzy sets, which is a measure. The cocept of cardiality of a fuzzy set is a extesio of the cout of elemets of a crisp set. A simple way of extedig the cocept of cardiality was suggested by Deluca ad Termii 4. This cocept is related to the otio of the probability measure of a fuzzy set itroduced by Zadeh 8 ad is termed as the sigma cout or the o-fuzzy cardiality of a set. Accordig to fuzzy set theory, the o-membership value of a elemet is oe s complemet of its membership value. However, i practical cases it is observed that this happes to be a serious costrait. So, Ataassov 1 itroduced the otio of ituitioistic fuzzy sets as a geeralisatio of the cocept of fuzzy sets which does ot have the deficiecy metioed above. Ulike, the cardiality of a fuzzy set (4,6,8) there are o defiitios of the cardiality of a ituitioistic fuzzy set i the literature. I this paper we itroduce the sigma cout as a extesio of the otio of the correspodig otio for fuzzy sets ad establish may properties. Also, we itroduce the otio of relative sigma cout ad establish some properties. Fially we illustrate the geeratio of quatificatio rules. Correspodig author. addresses: tripathybk@vit.ac.i (B. K. Tripathy), sp.jea08@gmail.com (S. P. Jea) ad r.swapa.ghosh@gmail.com (S. K. Ghosh)

2 124 B. K. Tripathy et al. / A Ituitioistic Fuzzy cout... 2 Defiitios ad Notatios I this sectio we shall provide some defiitios ad otatios to be used i this paper. First we itroduce the otio of a fuzzy set. Defiitio 2.1. Let X be a uiversal set. The a fuzzy set A o X is defied through a membership fuctio associated with A ad deoted by µ A as µ A : X 0, 1, (2.1) such that every xɛx is associated with its membership value µ A (x) lyig i the iterval 0, 1. Clearly, the fuzzy set A is completely characterized by the set of poits {(x, µ A (x)) : xɛx}. Defiitio 2.2. For ay two fuzzy sets A ad B i X, we defie the relatioships betwee A ad B as A B iff µ A (x) µ B (x), xɛx (2.2) A B iff µ A (x) µ B (x), xɛx (2.3) B A iff A B (2.4) Defiitio 2.3. The uio of the two fuzzy sets A ad B is give by its membership fuctio µ A B (x) defied by µ A B (x) max{µ A (x), µ B (x)}, xɛx. (2.5) Defiitio 2.4. The itersectio of the two fuzzy sets A ad B is give by its membership fuctio µ A B (x) defied by µ A B (x) mi{µ A (x), µ B (x)}, xɛx. (2.6) Defiitio 2.5. The complemet Ā of the fuzzy set A with respect to uiversal set X is give by its membership fuctio µ Ā (x) defied by µ Ā (x) 1 µ A (x), xɛx. (2.7) Defiitio 2.6. Let X be a uiversal set. A itiitioistic fuzzy set or IFS A o X is defied through two fuctios µ A ad ν A, called the membership ad o-membership fuctios of A defied as µ A : X 0, 1 ad ν A : X 0, 1 (2.8) such that every xɛx is associated with its membership value µ A (x) ad o-membership value ν A (x) such that 0 µ A (x) + ν A (x) 1. Defiitio 2.7. If A ad B are two IFSs of the set X, the A B iff xɛx, µ A (x) µ B (x) ad ν A (x) ν B (x) (2.9) A B iff B A (2.10) A B iff xɛx, µ A (x) µ B (x) ad ν A (x) ν B (x) (2.11) Ā { x, ν A (x), µ A (x) : xɛx} (2.12) A B { x, mi(µ A (x), µ B (x)), max(ν A (x), ν B (x)) : xɛx} (2.13) A B { x, max(µ A (x), µ B (x)), mi(ν A (x), ν B (x)) : xɛx} (2.14) A + B { x, µ A (x) + µ B (x) µ A (x) µ B (x), ν A (x) ν B (x) : xɛx} (2.15)

3 B. K. Tripathy et al. / A Ituitioistic Fuzzy cout A B { x, µ A (x) µ B (x), ν A (x) + ν B (x) ν A (x) ν B (x) : xɛx} (2.16) A { x, µ A (x), 1 µ A (x) : xɛx} (2.17) A { x, 1 ν A (x), ν A (x) : xɛx} (2.18) C(A) { x, K, L : xɛx}, where K max xɛx µ A(x) ad L mi xɛx ν A(x) (2.19) I(A) { x, k, l : xɛx}, where k mi xɛx µ A(x) ad l max xɛx ν A(x) (2.20) 3 Cardiality of Ituitioistic Fuzzy Sets I this sectio, we itroduce the cardiality of ituitioistic fuzzy sets ad establish some properties. 3.1 Defiitios The measure of fuzzy set is the form of its Σ cout (sigma cout) was itroduced by Deluca ad Termii 4 as a simple extesio of the cocept of cardiality of crisp sets. As metioed above Ituitioistic fuzzy sets have better modelig power tha those of fuzzy sets, by the way itroducig the hesitatio part. Here we defie the cardiality of a Ituitioistic fuzzy set by extedig the otio of Σ cout stated above. Also, we establish some of their properties, ad provide certai examples ad applicatio of these results. Defiitio 3.1. A fuzzy set A o X to be fiite if µ A (x) 0 for oly a fiite umber of elemets of X. Defiitio 3.2. For ay fiite fuzzy set A o X, the sigma cout of A, deoted by Σ cout (A) is give by Σ cout (A) xɛx µ A (x) (3.21) Defiitio 3.3. For ay IFS A o X we defie cardiality of A (deoted by Σcout(A)) as Σ cout (A) µ A (x i ), 1 ν A (x i ) (3.22) Σ cout A, Σ cout A (3.23) It may be oted that whe A is a fuzzy set o X, ν A (x) 1 µ A (x), for all xɛx, so that Σ cout (A) µ A (x i ), µ A (x i ) which is the defiitio of Σ cout of a fuzzy set A defied above. µ A (x i ) 3.2 Properties of Σ cout We establish some properties of σ cout of IFSs i this sectio. Theorem 3.1. For ay two IFSs A ad B o X (i) Σ cout (A B) + Σ cout (A B) Σ cout (A) + Σ cout(b) (ii) Σ cout (A + B) + Σ cout (A B) Σ cout (A) + Σ cout(b)

4 126 B. K. Tripathy et al. / A Ituitioistic Fuzzy cout... Proof. We have ad So, Σ cout (A B) (µ A (x i ) µ B (x i )), 1 (ν A (x i ) ν B (x i )) (µ A (x i ) µ B (x i )), {(1 ν A (x i )) (1 ν B (x i ))} Σ cout (A B) (µ A (x i ) µ B (x i )), 1 (ν A (x i ) ν B (x i )) (µ A (x i ) µ B (x i )), {(1 ν A (x i )) (1 ν B (x i ))} Σ cout (A B) + Σ cout (A B) (µ A (x i ) + µ B (x i )), {(1 ν A (x i )) + (1 ν B (x i ))} The proof of (ii) is similar to that of (i). µ A (x i ), 1 ν A (x i ) + µ B (x i ), Σ cout (A) + Σ cout (B) Theorem 3.2. For ay two IFSs A ad B o X (i) Σ cout (A B) + Σ cout (A B) Σ cout (Ā) + Σ cout( B) (ii) Σ cout (A + B) + Σ cout (A B) Σ cout (Ā) + Σ cout( B) 1 ν B (x i ) Proof. A B { x, µ A (x) µ B (x), ν A (x) ν B (x) : xɛx} A B { x, ν A (x) ν B (x), µ A (x) µ B (x) : xɛx} So, similarly, Σ cout (A B) (ν A (x i ) ν B (x i )), 1 (µ A (x i ) µ B (x i )) (ν A (x i ) ν B (x i )), {(1 µ A (x i )) (1 µ B (x i ))} A B { x, µ A (x) µ B (x), ν A (x) ν B (x) : xɛx} A B { x, ν A (x) ν B (x), µ A (x) µ B (x) : xɛx} ad Σ cout (A B) (ν A (x i ) ν B (x i )), 1 (µ A (x i ) µ B (x i )) (ν A (x i ) ν B (x i )), {(1 µ A (x i )) (1 µ B (x i ))}

5 B. K. Tripathy et al. / A Ituitioistic Fuzzy cout Hece, Σ cout (A B) + Σ cout (A B) (ν A (x i ) + ν B (x i )), {(1 µ A (x i )) + (1 µ B (x i ))} ν A (x i ), 1 µ A (x i ) + ν B (x i ), 1 µ B (x i ) Σ cout (Ā) + Σ cout ( B) The proof of (ii) is similar to that of (i) Next, by usig the results of Ataassov 2,3, the followig properties of Σ cout of IFSs ca be obtaied. Theorem 3.3. For ay IFSs A, we have: (i) Σ cout A Σ cout ( Ā) (ii) Σ cout A Σ cout ( Ā) (iii) Σ cout A Σ cout A (iv) Σ cout A Σ cout A (v) Σ cout A Σ cout A (vi) Σ cout A Σ cout A (vii) Σ cout Ā Σ cout A (viii) Σ cout Ā Σ cout A (ix) Σ cout Ā Σ cout A It may be oted that from the defiitio of Σ cout of a IFS, it ca be obtaied directly that if A B the it is ot always true that Σ cout A Σ cout B. Also (x) Σcout A Σcout A Proof. We have ad Σ cout A µ A (x i ), 1 (1 µ A (x i )) Σ cout A (1 ν A (x i )), (1 ν A (x i )) µ A (x i ) (1 ν A (x i )) Also, by the defiitio of a IFS, µ A (x i ) 1 ν A (x i ), i 1, 2,...,. So the claim follows. Theorem 3.4. For ay two IFSs A ad B, (i) Σcout (A B) Σcout ( A B) (ii) Σcout (A B) Σcout( A B) (iii) Σcout (A B) Σcout( A B) (iv) Σcout (A B) Σcout( A B) (v) Σcout (A B) Σcout(Ā B) (vi) Σcout (A B) Σcout(Ā B) Theorem 3.5. For ay two IFSs A ad B o X, (i) Σcout (A B) + Σcout (A B) Σcout A + Σcout B (ii) Σcout (A B) + Σcout (A B) Σcout A + Σcout B (iii) Σcout (A + B) + Σcout (A B) Σcout A + Σcout B (iv) Σcout (A + B) + Σcout (A B) Σcout A + Σcout B

6 128 B. K. Tripathy et al. / A Ituitioistic Fuzzy cout... Proof. (i)σcout (A B) Σcout ( A B) ad Σcout (A B) Σcout ( A B) so, Σ cout (A B) + Σ cout (A B) Σ cout ( A B) + Σ cout ( A B) Σ cout A + Σ cout B Similarly (ii) ca be established. (iii) Σcout (A + B) + Σcout (A B) Proof. (µ A (x i ) + µ B (x i ) µ A (x i ) µ B (x i )) + µ A (x i ) µ B (x i ) µ A (x i ) + µ B (x i ) Σ cout A + Σ cout B Similarly (iv) ca be established. Note By usig Theorems 3.1, 3.2 ad 3.5, we have the followig (i) Σcout(A B) + ΣcoutA B Σcout(A + B) + Σcout(A B) Σ cout A + Σ cout B (ii) Σcout(A B) + ΣcoutA B Σcout(A + B) + Σcout(A B) Σ cout Ā + Σ cout B (iii) Σcout (A B) + Σcout (A B) Σcout (A + B) + Σcout (A B) Σ cout A + Σ cout B (iv) Σcout (A B) + Σcout (A B) Σcout (A + B) + Σcout (A B) 3.3 Relative Σ cout Σ cout A + Σ cout B The otio of relative Σ cout for fuzzy sets has bee itroduced by Zadeh 9. Defiitio 3.4. If A ad B are two fuzzy sets, the we defie the relative sigma cout of A with respect to B as rel Σcout (A/B) (Σcout(A B)) (Σcout(B)) 1, if A ad b are two IFSs, the Σ cout (A B) µ A (x i ) µ b (x i ), 1 (ν A (x i ) ν B (x i )) 1 (Σ cout (B)) 1 1 µ B (x i ), 1 ν B (x i ) (1 ν B(x i )), 1 µ B(x i ) Cosequetly, (Σ cout (A B)) (Σ cout (B)) 1 µ A(x i ) µ B (x i ) (1 ν, B(x i )) µ A(x i ) µ B (x i ) (1 ν, B(x i )) {1 ν A(x i )) (1 ν B (x i ))} µ B(x i ) 1 (ν A(x i ) ν B (x i )) µ B(x i )

7 B. K. Tripathy et al. / A Ituitioistic Fuzzy cout It may be oted that the right had expressio of the above iterval may be greater tha 1. For example, takig X {x 1, x 2 } ad A ad B two IFSs over X defied by A {(.8,.1)/x 1, (.2,.6)/x 2 }, B {(.6,.2)/x 1, (.3,.6)/x 2 }. The (1 ν A (x 1 )) (1 ν B (x 1 )) + (1 ν A (x 2 )) (1 ν B (x 2 )) I view of the above remark, we defie rel Σ cout (A/B) for ituitioistic fuzzy sets as relσcout(a/b) µ ( A(x i ) µ B (x i ) (1 ν, mi 1, B(x i )) {(1 ν ) A(x i )) (1 ν B (x i ))} µ B(x i ) Some Pathological Cases Case I: Suppose A ad B are fuzzy sets. The A A, B B, 1 ν A µ A ad 1 ν B µ B. So relσcout(a/b) µ A(x i ) µ B (x i ) (1 ν B(x i )) which is same as the Prop (A/B) itroduced by Zadeh. Case II: Suppose A is a IFS ad B is a fuzzy set. The,1 ν B µ B. So that relσcout(a/b) (µ A(x i ) µ B (x i )) µ, ((1 ν A(x i )) µ B (x i ) B(x i ) µ B(x i ) 1 µ B(x i ) (µ A (x i ) µ B (x i )), ((1 ν A (x i ) µ B (x i )) Also, i this case the right had limit of the iterval is less tha or equal to 1. So, we eed ot impost this additioal restrictio. Case III: If a is a fuzzy set ad B is a crisp set, the rel Σcout(A/B) µ ( A(x i ), mi 1, µ ) A(x i ) Card(B) Card(B) I particular whe B X {x 1, x 2,... x }, we get rel Σcout(A/B) µ ( A(x i ), mi 1, µ ) A(x i ) µ A(x i ) 1 µ A (x i ) 4 Some Applicatios Defiitio 4.1. e 1, e 2, where e 1 Let A ad B be two IFSs o X. The the rel cout(a/b) is defied by the iterval (µ A(x i ) µ B (x i )) (1 ν B(x i )) ( mi 1, ((1 ν ) A(x i )) (1 ν B (x i )) µ B(x i ) Here e 1 idicates the miimum amout of similarity betwee A ad B ad e 2 idicates the maximum amout of similarity betwee a ad B. Clearly, rel Σcout(A/B) 0, 1 ad rel Σcout(A/B) relσcout(b/a) i geeral. relσcout(a/a) µ A(x i ). (1 ν A(x i )), 1 Defiitio 4.2. For a give class {A i }iɛλ of IFSs o X, the IFS S o X is said to be the super IFS if S {< x, µ s (x), ν s (x) >: xɛx} where µ S (x) sup µ Ai (x) ad ν S (x) if ν A i (x). iɛλ iɛλ Defiitio 4.3. Let A ad B two IFSs o X. The we say A domiates B if mid value(rel Σ cout(a/s)) mid value (rel Σ cout (B/S))

8 130 B. K. Tripathy et al. / A Ituitioistic Fuzzy cout... C 1 C 2 C 3 C 4 C 5 C 6 A (.2,.7) (.5,.2) (.8,.1) (.6,.3) (.4,.5) (.3,.6) B (.6,.2) (.2,.7) (.7,.3) (.8,.2) (.5,.3) (.9,.1) C (.2,.7) (.4,.5) (.8,.2) (.9,.1) (.6,.3) (.5,.2) D (.5,.4) (.3,.5) (.6,.3) (.5,.3) (.7,.2) (.9,.0) S (.6,.2) (.5,.2) (.8,.1) (.9,.1) (.7,.2) (.9,.0) 4.1 Case Studies Case Study 1: Cosider the problem of gradatio of studets of a class. The Characteristics, which are to determie the gradatio, may be some characteristics as Skill Kowledge Disciplie i the school Puctually Efficiecy i extracurricular activities Age A selector may have to use the above characteristics ad make their evaluatio for each studet i a class, cosiderig all the iformatio. The gradatio list ca be prepared basig upo the evaluatio ad some techique. We may use the techique of domiace as defied i defiitio 4.3 as the factor of gradatio. To make a case study, we assume that the umber of characteristics be six. O the basis of these six characteristics which we deote by C 1, C 2, C 3, C 4, C 5 ad C 6, suppose there are four studets with the characteristics as metioed above i the form of a matrix: The super IFS S will be give above i the form of matrix: relσcout(a/s) (µ ( A(x i ) µ S (x i )) 1, (1 ν, mi ((1 ν ) A(x i )) (1 ν B (x i ))) s(x i )) µ S(x i ) relσcout(b/s) relσcout(c/s) relσcout(d/s) ( , mi 1, 3.6 ) , 9.54, ( , mi 1, 4.2 ) , 21.71, ( ) , mi , , 10.65,.9 11 ( , mi 1, 4.3 ) , 43.67, The mid values of relσcout(a/s), relσcout(b/s), relσcout(c/s) ad relσcout(d/s) are.68,.83,.75,.825 respectively. So, the gradig is B, D, C, A. Defiitio 4.4. Let A be a IFS o X {x 1, x 2,..., x }. The depth of A deoted by depth (A) is give by depth(a), ΣcoutA, a 1, a 2 where a 1 µ A (x i ) ad a 2 (1 ν A (x i ))) a 2, a 1 Clearly, depth(x) 0 ad depth (φ).

9 B. K. Tripathy et al. / A Ituitioistic Fuzzy cout Defiitio 4.5. Let A 1 ad A 2 be two IFSs over X, the we say A 2 is a better represetative of X tha A 1 deoted by A 2 A 1, if ad oly if where a, b is give by max( a, b ). depth(a 2 ) < depth(a 1 ) Usig the above defiitios a gradig of IFSs defied over a set X ca be made. The orderig beig the A i comes higher i the order the A k if A k is a better represetative of X tha A i. We explai this by a case study. Case Study 2: Cosider four IFSs A 1, A 2, A 3 ad A 4 defied over the fiite set X {x 1, x 2 } give by A 1 {(.5,.4)/x 1, (.2,.8)/x 2 } A 2 {(.1,.8)/x 1, (.9, 0)/x 2 } A 3 {(.1,.9)/x 1, (0, 1)/x 2 } ad A 4 {(.2,.5)/x 1, (.1,.7)/x 2 } ad Here depth(a 1 ) 2.8, , depth(a 2 ) 2 1.2, 2.1.8, 1 1 depth(a 3 ) 2.1, , depth(a 4 ) 2.9, , So, A 2 A 1 A 4 A 3. Thus A 2 is the best represetative of x. 5 Quatificatio Rules If x is A be a propositio, the the propositio is modified by the modifier by m as ot, very, fairly etc. Hece the modifier propositio be x is ma. Similarly propositio may be quatified by ituitioistic fuzzy quatifiers such as usually, frequetly, most etc. Quatifiers are deoted by Q. So, Qx s are A s is a quatified propositio ad QA s are B s is kow as exteded quatified propositios. For example, most cars are fast is a quatified propositio, where most fast cars are dagerous is a exteded quatified propositio. The exteded quatified propositio as QA s are B s, where Q is a ituitioistic fuzzy quatifier with membership fuctio µ Q (x) ad the o-membership fuctio ν Q (x) ad the IFSs A ad B have membership ad o-membership fuctios with the same argumet o xɛu, (µ A (x), ν A (x)) ad(µ B (x), ν B (x)) correspodigly. We have to fid out the truth of the above quatified propositios. Let A ad B are two IFSs o a fiite uiverse of discourse U {x 1, x 2,..., x } the ΣcoutA µ A (x i ), (1 ν A (x i )) ΣcoutB µ B (x i ), (1 ν B (x i ))

10 132 B. K. Tripathy et al. / A Ituitioistic Fuzzy cout... I particular Σ cout X,, where each x i, i 1, 2,... has a membership value 1 ad omembership value 0. The truth value of the propositio i a fiite uiverse U is determied by truth (QAs are Bs) (µ Q (r), ν Q (r)), where the value of r is Σcout(A B) r relσcout(b/a) (µ ( A(x i ) µ B (x i )) (1 ν, mi 1, s(x i )) Σcout(A) ((1 ν A(x i )) (1 ν B (x i )) µ A(x i ) The meaig of the coefficiet r relσcout(b/a) is that it expresses the proportio of B i A. I particular case, whe A ad B are fuzzy sets istead of IFSs, the the propositio QA are B reduced to the cocept of Zadeh s sese. Also, i the case, Qxs are Bs that is, whe istead of a IFS A, we have a crisp set {x i } U, the truth value of Qxs are B be truth(qx s are B) (µ Q (r 0 ), ν Q (r 0 )) where r 0 relσcout(b/u) µ ( B(x i ), mi 1, (1 ν ) B(x i )) Example 5.1. Cosider the propositio most cars are fast. Assume that cars, fast ad most are defied as cars y {y 1, y 2, y 3 }, U {y 1, y 2, y 3 } ) cars B (.1,.8)/y 1 + (.6,.2)/y 2 + (.8,.2)/y 3 ad most Q, where 0 0 x.3; µ Q (x) 1 {1 + (2x 0.6) 2 } 1.3 x.7; 1.7 x. ad 1 0 x.4; ν Q (x) {1 + (2x 0.8) 2 } 1.4 x.8; 0.8 x. r 0 relσcout(b/u) µ ( B(x i ), mi 1, (1 ν ) B(x i )) ( 1.5 3, mi 1, 1.8 ).5,.6 3 mid value (r 0 ).55, which is the average of the degree of car speed. Now substitutig r 0.55 for x, we have µ Q (.55).2 ad ν Q (.55).53 The truth value depeds o how both the quatifiers Q(most) ad the set B(fast) are defied. Example 5.2. Let us cosider the more geeral propositio, Most fast cars are dagerous, usig the data i the above example for cars, fast ad most. I additio, let dagerous be defied as dagerous A (.2,.7)/x 1 + (.5,.4)/x 2 + (.6,.4)/x 3

11 B. K. Tripathy et al. / A Ituitioistic Fuzzy cout to defie r we have to calculate r relσcout(b/a) mid value (r) which represet the proportio of B i A. Fially, substitutig r for x, we have ( , mi 1, 1.4 ) ( , mi 1, 14 ) , µ Q (.9) 1 ad ν Q (.9) 0. 6 Coclusio I this paper a measure of cardiality of IFS, called Σcout which geeralizes the otio of Σcout of fuzzy sets itroduced 4 has bee put forth ad studied. May results ivolvig Σ cout of trasformed IFSs by usig modal operatios have bee established. A otio called relative Σ cout is defied ad as a applicatio, a case study is made. Ituitioistic fuzzy quatifiers are discussed ad illustrated by takig examples. Refereces Ataassov K. T., Ituitioistic fuzzy sets, Fuzzy Sets ad Systems., 20,(1986), Ataassov K. T., More o ituitioistic fuzzy sets, Fuzzy Sets ad Systems, 33,(1989), Ataassov K. T., New operatios defied over ituitioistic fuzzy sets, Fuzzy Sets ad Systems, 61,(1994), DeLuca A. ad Termii S., A defiitio of a o-probabilistic etropy i the settig of Fuzzy Set Theory, Iformatio ad Cotrol, 20,(1972), Klir G.J. ad Yua B., Fuzzy sets ad fuzzy logic, theory ad applicatios, Pretice - Hall of Idia, (1997). Tripathy B. K., Jea S. P. ad Ghosh S. K., A bag theoretic fuzzy cout ad cardiality of fuzzy sets, commuicated to Iteratioal Joural of Ucertai Systems. Zadeh L. A., Fuzzy sets, Iformatio ad Cotrol, 8 (1965), Zadeh L. A., Probability mesures of fuzzy evets, Joural of Mathematical Aalysis ad Applicatios, 23(1968), Zadeh L. A., Test-score sematics for atural laguages ad meaig represetatio via PRUF, Riege,B.(ed.), Empirical Sematics, Brockmeyer, bochum, Germay,(1982), Received: November 11, 2012; Accepted: July 06, 2013 UNIVERSITY PRESS

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