IOSR Joural of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 239-765X. Volume, Issue 2 Ver. VI (Mar - pr. 205), PP 0-05 www.iosrjourals.org Fuzzy Subspace, Balaced Fuzzy Set ad bsorbig Fuzzy Set i Fuzzy Vector Space M.Z.LM Dept.ofMathematics,MillatCollege,Darbhaga 846004.(BIHR),INDI bstract: I this paper, we have studied fuzzy subspace, balaced fuzzy set, ad absorbig fuzzy set over a fuzzy vector space. We eamie the properties of the balace fuzzy set, ad absorbig fuzzy set, ad established some idepedet results, uder the liear mappig from oe fuzzy vector space to aother fuzzy vector space. Keywords:Fuzzy vector space, Fuzzy balaced set, Fuzzy absorbig set. I. Itroductio: The cocept of fuzzy set was itroduced by Zadeh [6],ad the otio of fuzzy vector space was defied ad established by KTSRS,.K. ad LIU,D.B [2].Usig the defiitio of fuzzy vector space, balaced fuzzy set ad absorbig fuzzy set, over the fuzzy vector space. We established the result that uder the liear mappig, these balaced fuzzy set, ad absorbig fuzzy set, remais the same.. Fuzzy Vector Space Defiitio. : Let X be a vector space over K,where K is the space of real or comple umbers, the the vector space equipped with additio (+) ad scalar multiplicatio defied over the fuzzy set (o X ) as below is called a fuzzy vector space. dditio (+) : Let,..., be the fuzzy sets o vector space X, let f : X X, such that f (,..., )...,we defie... f,..., by the etesio priciple y sup,..., f,...,,... y f,... Obviously, whe sets,..., are ordiary sets,the gradatio fuctio used i the sum are take as characteristic fuctio of the set. Scalar multiplicatio (.) : If is a scalars ad B be a fuzzy set o X ad g : X X,such that g,the usig etesio priciple we defie B as B g B,where g B y y 0 sup {µ B ()}, if y holds yg y gb, if y,for ay sup, if y X i.e y B B y 0 y, if y B,for ay X THEOREM. :If E ad F are vector spaces over K,f is a liear mappig from E to F ad,b are fuzzy sets o E,the f B f f B f f, for all scalars α i.e f B f f B, where α, β, are scalars Proof: Proof is straight forward. Defiitio.2: If is a fuzzy set i a vector space E ad X, we defie + as + = {} +. DOI: 0.9790/5728-26005 www.iosrjourals.org Page
Fuzzy Subspace, Balaced Fuzzy Set ad bsorbig Fuzzy Set i Fuzzy Vector Space THEOREM.2 :If f : E E (vector space) such that f (y) = + y, the if B is a fuzzy set i E ad is a ordiary subset of E, the followig holds z B z B f B B B B Proof : Proof is straight forward. THEOREM.3: If,...,, are fuzzy sets i vector space E ad α,...,α are scalars... iff for,..., i E, we have... mi,..., Proof : Proof is obvious. 2. Fuzzy Subspace fuzzy set F i a vector space E is called fuzzy subspace of E if F + F F (ii)α F F, for every scalars α. THEOREM 2.:If F is a fuzzy set i a vector space E, the the followigs are equivalet (ii) (iii) F is a subspace of E For all scalars k,m, kf + mf F For all scalars k,m, ad all, y E k my mi, y F F F Proof : It is obvious THEOREM 2.2 :If E ad F are vector spaces over the same field ad f is a liear mappig from E to F ad is subspace of E. The f() is a subspace of F ad if B is a subspace of F. The f - (B) is a subspace of E. Proof : Let k, m, be scalars ad f is a liear mappig from E to F, the for ay fuzzy set i E f k m f kf mf f k f m s k + m, sice is a subspace. f() is a subspace of F k my lso, f B B f k my k my f B B kf mf y k my mi, f B B f B f y k my mi, y f B f B f B f B i.e, is a subspace of F DOI: 0.9790/5728-26005 www.iosrjourals.org 2 Page, sice f is a liear mappig, as B is a subspace. THEOREM 2.3: If, B, are fuzzy subspace of E ad K is a scalars. The + B ad K are fuzzy subspaces. Proof: Proof is obvious. 3. Balaced Fuzzy Set fuzzy set i a vector space E is said to be balaced if α, for all scalars α with I I. THEOREM 3.: Let be a fuzzy set i a vector space E. The the followig assertios are equivalet. is balaced (ii) (iii), for all scalars α with I α I For each α [0,], the ordiary set α give by E :, is balaced Proof : (ii) Suppose is balaced i.e α, for all scalars α with I α I.
Fuzzy Subspace, Balaced Fuzzy Set ad bsorbig Fuzzy Set i Fuzzy Vector Space i.e i e., for all scalars α with I α I, takig α for, where α 0.., for all scalars α, with I α I ad E If α = 0,from 0 0 sup y Suppose, (ii) (iii) i. e E Let : Now, t t Sice t, where α = 0 ye, for all α with I α I ad E :, [0,], with I t I, let, with I α I, whe I t I t, is balaced, with I t I (iii) Let E, ad let k Now,where I k I k, where y : y k k : k.,sice k, k i.e k, as k k k k, is balaced., for all scalars k with I k I, ad E is balaced THEOREM 3.2: Let E, F be vector spaces over k ad let f: E F be a liear mappig. If is balaced fuzzy set i E. The f() is balaced fuzzy set i F. Similarly f - (B) is balaced fuzzy set i Wheever B is balaced fuzzy set i F. Proof : Let E, F be vector spaces over k ad f : E F be a liear mappig. Suppose is balaced fuzzy set i E. Now α.f() = f (α) f(), for all scalars α with I α I i.e α. f() f(),hece f() is balaced [ α ] gai suppose B is a balaced fuzzy set i F α B B, for all scalars α with I α I Now, let M = α f - (B), therefore, f(m) = f (α f - (B)) = α f (f - (B)) α B B M f - (B), hece α f - (B) f - (B), therefore f - (B), is balaced fuzzy set i E. THEOREM 3.3: If, B are balaced fuzzy sets i a vector space E over K. The + B is balaced fuzzy set i E. Proof : Let,B are balaced fuzzy sets i E. Therefore α, ad α B B, for all scalars α with I α I, Now α ( + B) = α + α B + B, hece + B is balaced fuzzy set i E. THEOREM 3.4 :If { i } i I,is a family of balaced fuzzy sets i vector spaces E. The = i, is balaced fuzzy set i E Proof : Sice { i } i I, is a family of balaced fuzzy sets i E α i i, for all scalars α with I α I i.e DOI: 0.9790/5728-26005 www.iosrjourals.org 3 Page
that is, i i Fuzzy Subspace, Balaced Fuzzy Set ad bsorbig Fuzzy Set i Fuzzy Vector Space, for all scalars α with I α I Now let, = i y if i y ii, for all y E if i ii, take y = α if i ii, for all scalars α with I α I, ad E i ii, is balaced fuzzy set i E 4. bsorbig Fuzzy Set E k Defiitio : fuzzy set i a vector space E is said to be absorbig if k0. THEOREM 4. : For a fuzzy subset of of a vector space E, the followig are equivalet. is absorbig k (ii) For each E, sup k0 E : (iii) For each α with 0 α,the ordiary set Proof : (ii) ' E k Let ' E k 0 K ' K ' 0 sup E k ' 0 k0, for all E k ', Now put sup k (ii) (iii), Suppose for each ϵ E sup t,let t0 t ty y : t Now, k ' k E :, 0 α sup 0 t0. Hece for every there is t,such that t, where 0 α, Let = t y, the y Hece t, for some t > 0, α is absorbig (iii) Suppose for 0 α, The for each E, t α, for some, t > 0 The = t y, where, μ (y) α sup t t0 t0 t sup t t0 t0 E t y sup : t t0 [ 0 α ], is absorbig, is absorbig DOI: 0.9790/5728-26005 www.iosrjourals.org 4 Page
Fuzzy Subspace, Balaced Fuzzy Set ad bsorbig Fuzzy Set i Fuzzy Vector Space t t0, t0 t E,Therefore is absorbig. THEOREM 4.2: Let E, F be a vector space over K ad f : E F, be a liear mappig. If is a absorbig fuzzy set i F. The f - () is a absorbig fuzzy set i E. Proof : Suppose is a absorbig fuzzy set i F. Let E sup sup k f k, f f f k0 k0 k kf k k sup sup f k0 0 f k0 [ by etesio priciple, [ f is liear ] sup, as f() F, ad is absorbig fuzzy set i F f - () is a absorbig fuzzy set i E. E ] Refereces []. KNDEL, ad BYTT,W.J.(978). Fuzzy sets, Fuzzy algebra ad Fuzzy statistics. Proceedig of ILEE, Vol.66, ad No.2 PP.69-639. [2]. KTSRS,.K. ad LIU,D.B.(977). Fuzzy vector spaces ad Fuzzy topological vector spaces. J.Math.al.ppl.58, PP. 35-46. [3]. LKE,J.(976). Sets, Fuzzy sets, multisets, ad Fuctios. Joural of Lodo Mathematical Society, 2. PP.323-326. [4]. NGUYEN,H.T.(978). ote o the etesio priciple for fuzzy sets. UCE/ERL MEMO M-6.UNIV.OF CLIFORNI, BERKELEY. [lso i J.Math.al.ppl.64, No.2, ad PP.369-380.] [5]. RUDIN,W. Fuctioal alysis,mcgraw-hill Book Compay, New York,973. [6]. ZDEH,L.. Fuzzy sets Iform.Cotrol,8.(965),PP. 338-353. [7]. ZDEH,L..(975). The cocept of a liguistic variable ad its applicatio to approimate Reasoig. Iform.Sci.8, (975), 99-249, 30-357, 9 (975),43-80. DOI: 0.9790/5728-26005 www.iosrjourals.org 5 Page