A NOTE ON FUZZY CLOSURE OF A FUZZY SET

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1 (JPMNT) Journal of Process Management New Technologes, Internatonal A NOTE ON FUZZY CLOSURE OF A FUZZY SET Bhmraj Basumatary Department of Mathematcal Scences, Bodoland Unversty, Kokrajhar, Assam, Inda, E-mal:brbasumatary14@gmal.com Abstract: In ths artcle fuzzy closure has been dscussed wth the help of extended defnton of fuzzy set on the assumpton that the unon of a fuzzy set and ts complement s unversal set and ntersecton of a fuzzy set and ts complement s empty set. Also we have dscussed some proposton of fuzzy closure wth the help of numercal example on the bass of extended defnton of fuzzy set. Keywords: Fuzzy membershp functon, Fuzzy reference functon, Fuzzy membershp value, Fuzzy closed set, Fuzzy closure. 1. Introducton Fuzzy set theory was dscovered by Zadeh [1] n Chang [2] ntroduced fuzzy topology. After the ntroducton of fuzzy sets and fuzzy topology, several researches were conducted on the generalzatons of the notons of fuzzy sets and fuzzy topology. The theory of fuzzy sets actually has been a generalzaton of the classcal theory of sets n the sense that the theory of sets should have been a specal case of the theory of fuzzy sets. But unfortunately t has been accepted that for fuzzy set A and ts complement A C, nether A A C s empty nor A A C s the unversal set. Whereas the operatons of unon and ntersecton of crsp sets are ndeed specal cases of the correspondng operaton of two fuzzy sets, they end up gvng pecular results whle defnng A A C and A A C. In ths regard Baruah [3, 4, 5,] has forwarded an extended defnton of fuzzy sets whch enable us to defne complement of fuzzy sets n a way that gve us A A C s empty and A A C s unversal set 2. Extended defnton of fuzzy set Baruah [3, 4, 5,] gave an extended defnton of complementaton of fuzzy sets. Accordng to Baruah [3, 4, 5] to defne a fuzzy set, two functons namely fuzzy membershp functon and fuzzy reference functon are necessary. Fuzzy membershp value s the dfference between fuzzy membershp functon and fuzzy reference functon. Let µ 1 (x) and µ 2 (x) be two functons such that 0 µ 2 (x) µ 1 (x) 1. For fuzzy number denoted by {x, µ 1 (x), µ 2 (x); xϵ U}, we call µ 1 (x) as fuzzy membershp functon and µ 2 (x) a reference functon such that (µ 1 (x) - µ 2 (x) ) s the fuzzy membershp value. Let A = {x, µ 1 (x), µ 2 (x); xϵ U} and B ={x, µ 3 (x), µ 4 (x); xϵ U} be two fuzzy sets. Then we would have accordng to our way: A B= {x, mn ( µ 1 (x), µ 3 (x)), max (µ 2 (x), µ 4 (x) ); xϵ U} and A B = {x, max ( µ 1 (x), µ 3 (x)), mn (µ 2 (x), µ 4 (x) ); xϵ U}. Two fuzzy sets C= {x, µ C (x); xϵ U } and D= {x, µ D (x); xϵ U } n the usual defnton would be expressed as C= {x, µ C (x), 0; xϵ U } and D= {x, µ D (x), 0; xϵ U }. Now for two fuzzy sets A={x, µ(x), 0; xϵx} and B={x, 1, µ(x); xϵx} are defned over the same unverse X. Then we would have A B= {x, mn ( µ(x), 1), max ( µ(x), µ(x) ; xϵ U} = {x, µ(x), µ(x) ; xϵ U}, 35

2 (JPMNT) Journal of Process Management New Technologes, Internatonal Vol. 3, No. 4, whch s nothng but the null set. In other words, B defned above s nothng but A C complement n the classcal sense of the set theory. That s f we defne fuzzy sets A C ={x, 1, µ(x); xϵx}, t can be seen that t s nothng but the complement of the fuzzy sets A={x, µ(x), 0 ; xϵx }. Also we have A B ={x, max ( µ(x), 1), mn ( 0, µ(x) ); xϵu}={x, 1, 0; xϵu}, whch s nothng but the unversal set. We therefore conclude that f we express the complement of a fuzzy set A={x, µ(x), 0; xϵx} as A C ={x, 1, µ(x); xϵx}, then we get 1. A A C = the null set ф, and 2. A A C = the unversal set X. Ths would enable us to establsh that the fuzzy sets do form a feld f we defne complementaton n our way. 3. Basc operatons Let A={x, µ 1 (x), µ 2 (x); xϵu} and B={x, µ 3 (x), µ 4 (x); xϵu} be two fuzzy sets defned over the same unverse U.. A B ff µ 1 (x) µ 3 (x) and µ 4 (x) µ 2 (x) for all xϵu.. A B ={x, max(µ 1 (x), µ 3 (x)), mn(µ 2 (x), µ 4 (x))} for all xϵu.. A B ={x, mn(µ 1 (x), µ 3 (x)), max(µ 2 (x), µ 4 (x))} for all xϵu. If for some xϵu, mn(µ 1 (x), µ 3 (x)) max(µ 2 (x), µ 4 (x))}, then our concluson wll be A B= v. xϵu} A C ={x, µ 1 (x), µ 2 (x); xϵu} C ={x, µ 2 (x), 0; xϵu} {x, 1, µ 1 (x); v. If D = {x, µ(x), 0; xϵu} then D C ={x, 1, µ(x); xϵu} for all xϵu. Proposton 1: For fuzzy sets A, B, C over the same unverse X, we have the followng proposton. A B, B C A C. A B A, A B B. A A B, B A B v. A B A B=A v. A B A B=B Proposton 2: Let τ= {A r : rϵi} be a collecton of fuzzy sets over the same unverse U. Then... { v. { A ={x, max(µ 1 ), mn(µ 2 ); xϵu} A ={x, mn(µ 1 ), max(µ 2 ); xϵu} A } C = A } C = A C {A } C 4. Fuzzy Topology Defnton: A fuzzy topology on a nonempty set X s a famly τ of fuzzy set n X satsfyng the followng axoms (T1) 0 X, 1 X τ (T2) G 1 G 2 τ, for any G 1,G 2 τ (T3) G τ, for any arbtrary famly {G : G τ, I}. In ths case the par (X,τ) s called a fuzzy topologcal space and any fuzzy set n τ s known as fuzzy open set n X and clearly every element of τ C s sad to be closed set. 36

3 (JPMNT) Journal of Process Management New Technologes, Internatonal Here 1 x ={x, 1, 0; xϵx} and 0 X ={x, µ(x), µ(x)}, where µ(x) s taken from reference functon compared to relatve fuzzy set. Closure of fuzzy set: Let (X, τ) be fuzzy topology and A={x, µ(x), γ(x); xϵx} be fuzzy set n X. Then fuzzy closure of A are defned by Cl(A)= {G: G s fuzzy closed set n X and A G} set n X s fuzzy topology on X. n fuzzy set C the membershp value of a s 0.1 and the membershp value of b s 0.2. Proposton 3: Let (X, τ) be fuzzy topology and C be fuzzy set n X, then cl(0 X )=0 X. Example : set n X s fuzzy topology on X. Clearly we can show cl(0 X )=0 X. Proposton 4: Let (X, τ) be fuzzy topology and C be fuzzy set n X, then C cl(c). Example : set n X s fuzzy topology on X. n fuzzy set C the membershp value of a s 0.1 and the membershp value of b s 0.2. From ths example t s clear C cl(c). Proposton 5. Let (X, τ) be fuzzy topology and C, D be any two fuzzy sets n X, then Cl(C D)= Cl(C) Cl(D). Proposton3. Let (X, τ) be fuzzy topology and C, D be any two fuzzy sets n X, then f C D cl (C) cl (D). set n X s fuzzy topology on X. n fuzzy set C the membershp value of a s 0.1 and the membershp value of b s

4 (JPMNT) Journal of Process Management New Technologes, Internatonal Vol. 3, No. 4, D={{a, 0.6, 0.3}, {b, 0.5, 0.2}}, where n fuzzy set C the membershp value of a s 0.3 and the membershp value of b s 0.3. Clearly from our defnton of fuzzy set C D. Now cl(c)= {{a, 1, 0.4}, {b, 1, 0.3}} and cl(d)= {{a, 1, 0.2}, {b, 1, 0.1}}, whch shows that cl (C) cl (D). Hence C D cl (C) cl (D). Proposton 6. Let (X, τ) be fuzzy topology and C be any fuzzy sets n X, then cl(cl(c))=cl(c). set n X s fuzzy topology on X. n fuzzy set C the membershp value of a s 0.1 and the membershp value of b s 0.2. Now f we consder cl(c)=d, then cl(d)={{a, 1, 0.4}, {b, 1, 0.3} }=cl(c). Hence cl(cl(c))=cl(c). Proposton 7. Let (X,τ) be fuzzy topology and C, D be any two fuzzy sets n X, then Cl(C D)= Cl(C) Cl(D). set n X s fuzzy topology on X. n fuzzy set C the membershp value of a s 0.1 and the membershp value of b s 0.2. D={{a, 0.6, 0.3}, {b, 0.5, 0.2}}, where n fuzzy set C the membershp value of a s 0.3 and the membershp value of b s 0.3. Now cl(c)= {{a, 1, 0.4}, {b, 1, 0.3}} and cl(d)= {{a, 1, 0.2}, {b, 1, 0.1}}. Also C D= {{a, 0.5, 0.4}, {b, 0.5, 0.3}} {{a, 0.6, 0.3}, {b, 0.5, 0.2}} ={{a, 0.6, 0.3}, {b, 0.5, 0.2}} So Cl( C D)= {{a, 1, 0.2}, {b, 1, 0.1}}. Cl(C) Cl(D)= {{a, 1, 0.4}, {b, 1, 0.3}} {{a, 1, 0.2}, {b, 1, 0.1}}. ={{a, 1, 0.2}, {b, 1, 0.1}}. Hence Cl(C D)= Cl(C) Cl(D). 5. Conclusons We have seen that, n artcle [5] f a fuzzy set s characterzed wth respect to a extended defnton of fuzzy set, we can defne the complement of a fuzzy set n ts actual perspectve. As for the defnton usng a reference functon, we are clear that the orgnal defnton s suffcent for fuzzy arthmetc. However, from orgnal defnton A A C =ф does not follow. From Baruah s [3, 4, 5] defnton of complement of a fuzzy set we can remove ths dffculty. 38

5 (JPMNT) Journal of Process Management New Technologes, Internatonal Smlar problem we have faced n fuzzy topology too. In ths artcle we have seen that we can defne fuzzy closure and ts propostons very ncely. Also we have gven numercal examples on the bass of extended defnton of fuzzy set. Hope our work wll gve some contrbuton n further study of fuzzy topology. References [1] Zadeh L. A.,Fuzzy sets, Informaton and Control, 8, (1965). [2] Chang C. L., Fuzzy Topologcal Space, Journal of Mathematcal Analyss and Applcaton 24, (1968) [3] Baruah H. K., Fuzzy Membershp wth respect to a Reference Functon, Jr. of Assam Sc. Socety, Vol- 40, No-3, 65-73(1999). [4] Baruah H. K.,Towards Formng a Feld of Fuzzy Sets, Int. Jr. of Energy, Informaton and Communcatons, Vol. 2, ssue 1, Feb [5] Baruah H. K., The Theory of Fuzzy Sets: Belef and Realtes, Int. Jr. of Energy, Informaton and Communcatons, Vol. 2, ssue 2, May 2011 [6] Neog T. J. and Sut D. K., Complement of an Extended Fuzzy Set, Int. Jr. of Computer Applcaton ( ), Vol. 29-No.3 Sep