Convex Fuzzy Set, Balanced Fuzzy Set, and Absolute Convex Fuzzy Set in a Fuzzy Vector Space

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1 IOSR Journal of Mathematcs (IOSR-JM) e-issn: , p-issn: X. Volume 2, Issue 2 Ver. VI (Mar. - pr. 206), PP Conve Fuzzy Set, alanced Fuzzy Set, and bsolute Conve Fuzzy Set n a Fuzzy Vector Space Rajesh kr. sngh, mtkr.rya 2 &M.Z.lam 3 P.G.Dept.ofMathematcs,College of Commerce,rts&Scence,Patna (har) Inda. bstract:in ths paper, we have studed the absolute conve fuzzy set over a fuzzy vector space. We eamne the propertes of absolute conve fuzzy set, and establshed some ndependent results under the lnear mappng from one vector space to another one. Keywords : Fuzzy vector space, Fuzzy subspace, conve fuzzy set, alanced fuzzy set, and bsolute conve fuzzy set. I. Introducton The concept of fuzzy set was ntroduced by Zadeh [6], and the noton of fuzzy vector space was defned and establshed by KTSRS,.K and LIU,D. [2]. Usng the defnton of fuzzy vector space, balanced fuzzy set and absolute conve fuzzy set over a fuzzy vector space, we establshed the elementary propertes of absolute conve fuzzy set over a fuzzy vector space, usng the lnear mappng from one space to another one. II. Prelmnares FUZZY VECTOR SPCE Defnton 2. : Let X be a vector space over K,where K s the space of real or comple numbers, then the vector space equpped wth addton (+) and scalar multplcaton defned over the fuzzy set (on X ) as below s called a fuzzy vector space. n ddton (+) : Let,..., n be the fuzzy sets on vector space X, let f : X X, such that f (,..., )...,we defne n n n n... f,..., by the etenson prncple y sup,..., f,..., n,... n y f,... n n n Obvously, when sets,..., n as characterstc functon of the set. Scalar multplcaton (.) : If s a scalars and be a fuzzy set on X and g : X g,then usng etenson prncple we defne g y sup {µ ()}, f y yg y DOI: / Page are ordnary sets, the gradaton functon used n the sum are taken holds y 0 g, f y α, foranyεx y sup, f y X as g,where X,such that y.e y 0, f y, for any THEOREM 2. :If E and F are vector spaces over K,f s a lnear mappng from E to F and, are fuzzy sets on E, then Research scholar, P.G. Dept. of Mathematcs, College of Commerce, rts & Scence, Patna. 2 Research scholar, P.G.Dept. of Mathematcs, College of Commerce, rts & Scence, Patna. 3 ssocate Professor, P.G.Dept. of Mathematcs, College of Commerce, rts & Scence, Patna.

2 Conve Fuzzy Set, alanced Fuzzy Set, nd bsolute Conve Fuzzy Set In Fuzzy Vector Space f f f () f f () f f f.e Proof: Proof s straght forward., for all scalars α, where α, β, are scalars Defnton 2.2: If s a fuzzy set n a vector space E and X, we defne + as + = {} +. THEOREM 2.2: If f : E E (vector space) such that f (y) = + y, then f s a fuzzy set n E and s an ordnary subset of E, the followng holds z z f Proof : Proof s straght forward. THEOREM 2.3: If,..., n, are fuzzy sets n vector space E and α,...,α n are scalars... nn, ffor all,, n, ne,we have... mn,..., n n n n Proof : Proof s obvous. III. Fuzzy Subspace Defnton 3.: fuzzy set F n a vector space E s called fuzzy subspace of E f () F + F F () α F F, for every scalars α. THEOREM 3.:If F s a fuzzy set n a vector space E, then the followngs are equvalent () F s a subspace of E () For all scalars k,m, k F + m F F () For all scalars k,m, and all, y E k my mn, y F F F Proof : It s obvous THEOREM 3.2 :If E and F are vector spaces over the same feld and f s a lnear mappng from E to F and s subspace of E. Then f() s a subspace of F and f s a subspace of F. Then f - () s a subspace of E. Proof : Let k, m, be scalars and f s a lnear mappng from E to F, then for any fuzzy set n E f k m f kf mf f k f m s k + m, snce s a subspace. f() s a subspace of F lso, k my f f k my mn, k my f kf mf y k my f f y f k my mn, y f f f.ef - (), s a subspace of F DOI: / Page, snce f s a lnear mappng, as s a subspace. THEOREM 3.3 : If,, are fuzzy subspace of E and K s a scalars. Then + and K are fuzzy subspaces. Proof: Proof s obvous.

3 Conve Fuzzy Set, alanced Fuzzy Set, nd bsolute Conve Fuzzy Set In Fuzzy Vector Space IV. Conve Fuzzy Set Defnton 4.: fuzzy set n a vector space E s sad to be conve f for all αε [0,], α + α. THEOREM 4. :Let be a fuzzy set n a vector space E. Then the followng assertons are equvalent () s conve y mn, y, for all, y E, and for all αε [0,], () () For each αε [0,], the crsp set E Proof s obvous. :, s conve V. alanced Fuzzy Set Defnton 5.: fuzzy set n a vector space E s sad to be balanced f α, for all scalars α wth I α I. THEOREM 5. :Let be a fuzzy set n a vector space E. Then the followng assertons are equvalent. () s balanced () (), for all scalars α wth I α I For each α [0,], the ordnary set α gven by E :, s balanced Proof: () () Suppose s balanced.e α, for all scalars α wth I α I..e e. If α = 0,from (), for all scalars α wth I α I, takng α for, where α 0.. (), for all scalars α,wth I α I and E y 0 0 sup Suppose, () (). e E Let : Now, t t Snce t, where α = 0 ye, for all α wth I α I and E :, [0,],wth I t I, let, wth I α I, when I t I t, s balanced, wth I t I () () Let E, and let k k y : y, where,where I k I DOI: / Page

4 Conve Fuzzy Set, alanced Fuzzy Set, nd bsolute Conve Fuzzy Set In Fuzzy Vector Space : Now k k.e k k k,snce k k, s balanced., k. k.e k, for all scalars k wth I k I, and E, as s balanced THEOREM 5.2 : Let E, F be vector spaces over k and let f: E F be a lnear mappng. If s balanced fuzzy set n E. Then f() s balanced fuzzy set n F. Smlarly f - () s balanced fuzzy set n E, whenever s balanced fuzzy set n F. Proof : Let E, F be vector spaces over k and f : E F be a lnear mappng. Suppose s balanced fuzzy set n E. Now α.f () = f (α) f(), for all scalars α wth I α I.e α f() f(),hence f() s balanced [ α ] gan suppose s a balanced fuzzy set n F α, for all scalars α wth I α I Now, let M = α f - (), therefore, f(m) = f (α f - ()) = α f (f - ()) α M f - (), hence α f - () f - (), therefore f - () s balanced fuzzy set n E. THEOREM 5.3 : If, are balanced fuzzy sets n a vector space E over K. Then + s balanced fuzzy set n E. Proof : Let, are balanced fuzzy sets n E. Therefore α, and α, for all scalars α wth I α I, Now α ( + ) = α + α +, hence + s balanced fuzzy set n E. THEOREM 5.4 :If { } I,s a famly of balanced fuzzy sets n vector spaces E. Then =, s balanced fuzzy set n E Proof : Snce { } I, s a famly of balanced fuzzy sets n E α, for all scalars α wth I α I that s, Now let, = y nf I y nf I, for all scalars α wth I α I, for all y E nf I, take y = α I, s balanced fuzzy set n E, for all scalars α wth I α I, and E VI. bsolute Conve Fuzzy Set Defnton 6.: fuzzy set n a vector space E s sad to be absolutely conve f t s both conve and balanced. THEOREM 6.:Let be a fuzzy set n a vector space E. Then the followng are equvalent () s absolutely conve (), for all scalars, wth () y mn, y (v) For each [0,] n E. DOI: / Page,for all, ye and all scalars,,the crsp set E : wth s absolutely conve fuzzy set

5 Conve Fuzzy Set, alanced Fuzzy Set, nd bsolute Conve Fuzzy Set In Fuzzy Vector Space Proof :( ) ( ) Let s absolutely conve fuzzy set n E.e s conve as well as balanced..(i) for all scalars wth nd.(ii) for all scalars wth 0 Now puttng n (II) we get 2 (III) 2 2 ' ' ' ' Now for all scalars, wth we have From (I) ' and ' ' 2 lso.(a) ' 2 ddng (a) we get ' '. from (III) ' ' Let and then and 2 2 for all scalars, wth ' and Ths follows from theorem 3. v Suppose y mn, y, wth We take [0,] and then y mn, y Let E ' DOI: / Page for all, y E and all scalars for all, y E wth [0,] and (b) : [0,] If and y and y y mn, y y.e.(c), from (a) & (b) y y s conve. gan puttng 0, 0 n y mn, y 0 mn, y.e. 0, for all E If we put y 0 n y mn, y mn, 0,

6 Conve Fuzzy Set, alanced Fuzzy Set, nd bsolute Conve Fuzzy Set In Fuzzy Vector Space, for all and for all scalars wth Hence s balanced..e. s conve as well as balanced, mples that s absolutely conve. v Snce : E, s conve for every [0,] by theorem 4.. gan s balanced for every [0,] theorem 5.. Hence s absolutely conve.. Therefore fuzzy set s conve. Therefore fuzzy set s balanced by Theorem 6.2 :Every fuzzy subspace F of a vector space E s absolutely conve (.e. conve as well as balanced ) Proof : Suppose F s a fuzzy subspace F F F nd F F for all scalars F F F F F F F, wth Then F s conve, snce F F, for all scalars wth F s balanced, therefore F s absolutely conve as t s conve as well as balanced. Theorem 6.3 :If, are absolutely conve fuzzy sets n a vector space E. Then + s absolutely conve fuzzy sets n a vector space E Proof : Let, are absolutely conve fuzzy sets n a vector space E.e., are conve as well as balanced fuzzy sets n a vector space E Snce, are conve fuzzy sets n E, where [0,] lso, for all scalars [0,] Now, s conve fuzzy set n a vector space E. lso,, are balanced fuzzy sets n a vector space E, for all scalars wth nd, for all scalars wth.e., s absolutely conve fuzzy sets n a vector space E, snce t s conve as well as balanced. Theorem 6.4 :If s a famly of absolutely conve fuzzy sets n a vector space E. Then I I s also absolutely conve fuzzy set n E DOI: / Page

7 Conve Fuzzy Set, alanced Fuzzy Set, nd bsolute Conve Fuzzy Set In Fuzzy Vector Space I I Then, for all [0,].e. y mn, y Now let,, Then y nf I y y nf y Proof : Snce s a famly of absolutely conve fuzzy sets n a vector space E. Ths means that s a famly of conve as well as balanced fuzzy sets n E. I Let be a famly of conve fuzzy sets n E I for all y E I y mn, y y nf mn, y mn nf,nf y Hence s conve fuzzy set n E. I lso s a famly of balanced fuzzy sets n E I, for all scalars wth.e. Now let y, for all scalars wth () y nf, for all y E I nf, take y I, for all scalars wth,and E From () nf I s balanced fuzzy set n E, hence s absolutely conve fuzzy set n E. Theorem 6.5 : Let E, F are fuzzy vector space over K and f : E F be a lnear mappng () If s absolutely conve fuzzy set n E. Then f s absolutely conve fuzzy set n F. () If s absolutely conve fuzzy set n F. Then f s an absolutely conve fuzzy set n E. Proof () :Let be absolute conve fuzzy set n E,.e. s conve as well as balanced fuzzy set n E. Let [0,] and be conve fuzzy set n E, then DOI: / Page f f f f Hence f s conve fuzzy set n F gan s balanced fuzzy set n E f f f, for all scalars wth f Therefore s balanced fuzzy set n F f s conve as well as balanced fuzzy set n F, hence I I f s absolutely conve fuzzy set n F. () let s absolute conve fuzzy set n a vector space F mples that s conve as well as balanced fuzzy set n F Snce s a conve fuzzy set n F and let [0,]

8 Conve Fuzzy Set, alanced Fuzzy Set, nd bsolute Conve Fuzzy Set In Fuzzy Vector Space M f f Then f M Hence M f f M f f f s conve fuzzy set n E gan s a balanced fuzzy set n F for all scalars wth Now let M f f M f f M f.e. f f f s balanced fuzzy set n E Therefore, f s conve as well as balanced fuzzy set n E.e., f fuzzy set n E. s absolutely conve References []. KNDEL, and YTT,W.J.(978). Fuzzy sets, Fuzzy algebra and Fuzzy statstcs. Proceedng of ILEE, Vol.66, and No.2 PP [2]. KTSRS,.K. and LIU,D..(977). Fuzzy vector spaces and Fuzzy topologcal vector spaces. J.Math.nal.ppl.58, PP [3]. LKE,J.(976). Sets, Fuzzy sets, multsets, and Functons. Journal of London Mathematcal Socety, 2. PP [4]. NGUYEN,H.T.(978). note on the etenson prncple for fuzzy sets. UCE/ERL MEMO M-6.UNIV.OF CLIFORNI,ERKELEY. [lso n J.Math.nal.ppl.64,No.2,PP ] [5]. RUDIN,W. Functonalnalyss,MCGraw-Hll ook Company, New York,973. [6]. ZDEH,L.. Fuzzy sets Inform.Control,8.(965),PP [7]. ZDEH,L..(975). The concept of a lngustc varable and ts applcaton to appromate Reasonng. Inform.Sc.8, (975), , , 9 (975), DOI: / Page