A Study on Intuitionistic Fuzzy Number Group

Transcription

1 International Journal of Fuzzy Mathematics and Systems. ISSN Volume 2, Number 3 (2012), pp Research India Publications Study on Intuitionistic Fuzzy Number Group 1 Palanivelrajan M. and 2 Kaliraju K. 1 ssistant Professor, 2 Research Scholar, Ramanujan Research Centre, PG and Research Department of Mathematics, Government rts College (utonomous), Kumbakonam , TamilNadu, India. palanivelrajan1975@gmail.com, kaliraju64@yahoo.com bstract In this paper we made on attempt to study the algebraic properties of intuitionistic fuzzy number and introduce a new concept called intuitionistic fuzzy number group MS Subject Classification: 03E99, 05C25, Keywords: Intuitionistic fuzzy set, Intuitionistic fuzzy number, Intuitionistic fuzzy number group. Introduction Zadeh(1965) [5] introduced the concept of fuzzy set to handle the problem of uncertainty in the evaluation of many real life situations. tanassov.k(1986)[1] introduced Intuitionistic fuzzy set(ifs) as an etension to the fuzzy set, where the degree of membership denoting a non-belongingness to a set is eplicitly specified along with degree of membership of belongingness to the set. In 1987, Dubosis.D, Prade.H[ 3] discussed the mean value of a fuzzy number. tanassov.k.t(1989)[2] also discussed interval valued intuitionistic fuzzy set (IFS) as an another notable etension of fuzzy set. Grezegorzewski.P (2003)[4] defined two families of metrics and orders in space of Intuitionistic fuzzy numbers(ifn). In this paper we introduce the concept of Intuitionistic fuzzy number group (IFNG) and investigate some of its properties. Preliminaries In this section we state some definitions and properties related fuzzy set, Intuitionistic fuzzy set and intuitionistic fuzzy numbers.

2 270 Palanivelrajan M. and Kaliraju K. Let R be the set of all real numbers and F(R) all fuzzy subset defined on R. We denote F*(R) the set of all fuzzy number and also denote the set of all Intuitionistic fuzzy number by IF*(R). Let X be a universe of discourse, then a fuzzy set is defined as ={, μ( ) / X} which is characterized by a membership function μ ( ) : X [ 0,1 ], where μ ( ) denotes the degree of membership of the element to the set. n Intuitionistic fuzzy set (IFS) assigns to each element of the universe X a membership degree μ ( ) [ 0,1 ] and a non-membership degree ( ) [ 0,1 ] such that μ ( ) + ( ) 1. IFS is mathematically represented as {, μ ( ), ( ) / X}. The value π ( ) =1- μ ( ) - ( ) is called the degree of hesitancy or the Intuitionistic inde of to. 1. Let F(R) and is normal that is there eists R such that ( ) is called a fuzzy number. 2. Whenever λ [ 0,1 ] then λ {, ( ) λ} + λ λ, λ =1 then = is a closed interval denoted by + By the decomposition theorem of fuzzy sets a = λ aλ, a λ for every a F * ( R ) n Intuitionistic fuzzy subset = {, μ( ), ( ) / R} of the real line is called an intuitionistic fuzzy number if 0, 1 X such that μ ( 0) = 1 and μ ( 1) = 1 is if-conve (ie. its membership function μ is fuzzy conve and its nonmembership function is fuzzy concave). μ is upper semi continuous and is lower continuous. Supp = cl({ X : ( ) }) is bounded. Proposition Let and B be the Intuitionistic fuzzy number represented as =, μ ( ), ( ) / R, μ ( ), ( ) / R whose membership and { } and B ={ } B B

3 Study on Intuitionistic Fuzzy Number Group 271 non-membership values are represented as 0 if < a f ( ) if a1 < a2 μ( ) = 1 if a2 a3 g ( ) if a3 < a4 0 if a4 < 1 if < b1 h( ) if b1 < b2 ( ) = 0 if b2 b3 kb ( ) if b3 < b4 1 if b4 < nd 0 if < c1 fb( ) if c1 < c2 μb( ) = 1 if c2 c3 gb( ) if c3 < c4 0 if c4 < 1 if < d1 hb ( ) if d1 < d2 B( ) = 0 if d2 d3 kb ( ) if d3 < d4 1 if d4 < where a 1, a 2,a 3,a 4, b 1, b 2,b 3, b 4,c 1,c 2,c 3, c 4, d 1, d 2,d 3, d 4 R such that b 1 a 1 b 2 a 2 a 3 b 3 a 4 b 4, d 1 c 1 d 2 c 2 c 3 d 3 c 4 d 4 and the function f, g, h, k, f B, g B, h B, k B : R [ 0,1] ddition of two Intuitionistic fuzzy numbers and B Is defined as C=+B such that

4 272 Palanivelrajan M. and Kaliraju K. 0 if < a1+ c1 min { f ( ), fb( ) } if a1+ c1 < a2 + c2 μc ( ) = 1 if a2 + c2 a3+ c3 min { g ( ), gb( ) } if a3 + c3 < a4 + c4 0 if a4 + c4 < nd 1 if < b1+ d1 ma { h( ), hb( ) } if b1+ d1 < b2 + d2 C ( ) = 0 if b2 + d2 b3+ d3 ma { k( ), kb( ) } if b3 + d3< b4 + d4 1 if b4 + d4 < Multiplication of two Intuitionistic fuzzy numbers and B Is defined as C =.B such that μ ( ) =[min(a 1b 1,a 1 b 2,a 2 b 1,a 2 b 2 ),ma(a 1 b 1,a 1 b 2,a 2 b 1,a 2 b 2 ) ] and C C ( ) =[ma(a 1b 1,a 1 b 2,a 2 b 1,a 2 b 2 ),min(a 1 b 1,a 1 b 2,a 2 b 1,a 2 b 2 ) ], where =[a 1,a 2 ] and B=[b 1,b 2 ] Eample (b1 a1 b2 a2 a3 b3 a4 b4) 0 if < if 3 < 5 μ( ) = 1 if if 6 < 8 0 if 8< 1 if < if 2 < 4 ( ) = 0 if if 7 < 9 1 if 9< nd

5 Study on Intuitionistic Fuzzy Number Group if < if 3 < 5 μb( ) = 1 if if 6 < 8 0 if 8< 1 if < if 2 < 4 ( ) = 0 if if 7 < 9 1 if 9< ddition of two intuitionistic fuzzy number C=+B 0 if < if 6 < 10 μc ( ) = 1 if if 12 < 16 0 if 16< Such that 0 if < if 6 < 10 Such that μc ( ) = 1 if if 12 < 16 0 if 16< nd 1 if < if 4 < 8 C ( ) = 0 if if 14< 18 1 if 18< Proposition rithmetic operations on Intuitionistic fuzzy number satisfy some of its following * properties if abc,,, IF( R) we say that a * b= b * a (commutativity)

6 274 Palanivelrajan M. and Kaliraju K. ( a * b)* c= a *( b* c) (associativity) e* a = a* e= a (identity) a*( a) = ( a)* e= e (Inverse) Intuitionistic fuzzy number group rithmetic operations on a Intuitionistic fuzzy number forms a group. binary operation is a way of putting two things together. n algebraic system (IF * (R),*) be the set of all non-empty fuzzy numbers IF * (R) together with a composition * is called Intuitionistic fuzzy number group if the following aioms are satisfied. G1: (Closure law) The set IF * (R) is closed for the composition * * * that is * implies that a* b IF ( R) for all * * a IF R b IF R ( ), ( ) that G2:(ssociative law) The composition is associative that is ( a* b)* c= a*( b* c) for all abc,, IF*( R) G3: (Eistence of the identity) There eists an element e in IF *(R) Such that e* a = a* e= a for all a IF*( R) G4:(Eistence of inverse of each element) Each element in IF *(R) possesses an inverse in IF *(R) that is corresponding to each element a IF*( R) there eists an element b IF*( R) such that a* b= b* a= e (the element b is called the inverse of a and we write Theorem The identity element in a Intuitionistic fuzzy number group is unique. a 1 = b ). Proof Let IF*( R ) be a Intuitionistic fuzzy number group and if possible.let e and e ' be two identity elements in IF*( R ).Then ee. ' = e'. e= e' (since e ' is the identity) ee. ' = e'. e= e (since e is the identity ) Therefore e = e ' Hence the identity in IF*( R ) is unique.

7 Study on Intuitionistic Fuzzy Number Group 275 Theorem In a Intuitionistic fuzzy number group if a = a, b then it has a unique inverse. Proof Let IF*( R ) be a Intuitionistic fuzzy number group and e be the identity element in IF*( R ). Let a = a, b be an arbitrary element of IF*( R) and if possible let b and c be two inverse of a. Then ab. = ba. = e and ac. = ca. = e Now ba. b.( ac. ) = b. e= b = e implies that ( b. a). c = e. c = c and ac. But by the associative law ( ba. ). c= b.( ac. ) Therefore b = c showing that a has a unique inverse. = e implies that Hence each element of a Intuitionistic fuzzy number IF*( R ) has a unique inverse. Theorem The inverse of the inverse of an element of a Intuitionistic fuzzy number group is the element itself, if a = a, a. Proof Let IF*( R ) be a Intuitionistic fuzzy number group and e be the identity element in IF*( R ). Let a = a, a be an arbitrary element of IF*( R) Then a 1. a = e. Now a 1. a = e [ ] = [ ] 1 1 a a. a a. e ( a ). a. a = [ a ] 1 ea. = 1 [ a ] 1 Implies that a = [ a ] 1 1 Hence [ a ] 1 = a.

8 276 Palanivelrajan M. and Kaliraju K. Theorem Cancellation laws hold in a be a Intuitionistic fuzzy number group IF*( R ),for all abc,,, IF*( R) and if a = a, a then ab. = ac. implies that b= c (left cancellation law) and ba. = ca. implies that b = c(right cancellation law). Proof Let e be the identity element in IF*( R) and let a a, a = then ab. = ac. Implies that Implies that = 1 1 a. ab. a. ac. = 1 1 a. ab. a. ac. Implies that [ ] Implies that Implies that b gain ba. = c. a = 1 1 a. a. b a. a. c eb. = ec. = c Implies that [ ba. ]. a 1 = [ ca. ]. a 1 Implies that b. [ a. a 1 ] = c. [ a. a 1 ] Implies that be. = ce. Implies that b= c Theorem (Reversal law for inverse of the product) If, a = a, a, b= b, b ] in *( ) ab IF R *, ( ) ab. = b. a for all IF R.Then [ ] Proof Let e be the identity element in IF*( R) and let a a, a =, b= b, b ] Then [ ab, ] b, a ] = [ [ ] 1 ] = [ a[ 1 bb. 1 ] ]. a ab.. b. a = b. a (by associative law )

9 Study on Intuitionistic Fuzzy Number Group 277 = [ ae ] = e.. 1 a lso [ 1 1 b. a ] [ ab. ] 1 1 = [ [ ] 1 1 = [ [ ] ] = [ b 1. e ]. b = e b. a. a. b Thus [, ] ab b, a ] Hence[ ] ab. = b. a. b. a. a. b 1 1 =[ b. a ] [ ab. ] = e References [1] tanassov.k.t., Intuitionistic Fuzzy sets, Fuzzy sets and systems, 20(1986), [2] tanassov.k.t., Interval valued Intuitionistic Fuzzy Sets, Fuzzy sets and systems, 31(1989), [3] Dubosis D., Prade H., The Mean value of a Fuzzy Number, Fuzzy sets and systems 24, (1987) [4] Grzegorzewski.P., Distances and orderings in a family of Intuitionistic Fuzzy numbers, In Proceedings of the Third Conference on Fuzzy Logic and Technology (Eusflat03), pages , [5] Zadeh.L.., Fuzzy sets, Information Control, 8(1965),

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