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Vague Contra Generalized Pre Continuous Mappings Mary Margaret ISSN XXXX XXXX © 2016 IJESC Research Article Volume 6 Issue No. 8 Vague Contra Generalized Pre Continuous Mappings Mary Margaret. A1, AROCKIARANI. I2 Nirmala College for Women, Coimbatore, Tamil Nadu, India Abstract: The purpose of this paper is to introduce and investigate a new class of continuous mapping in vague topological spaces namely vague contra generalized pre continuous mapping, vague generalized pre homeomorphism and vague M -generalized pre homeomorphism. We also investigated some of its properties. Keywords: Vague topology, vague generalized pre homeomorphism, vague M-generalized pre homeomorphism, vague contra generalized pre continuous mappings. I. INTRODUCTION tA x,1 f A xis called the “vague value” of in and The theory of vague sets was first proposed by Gau and is denoted by VA x. Buehrer [6] as an extension of fuzzy set theory and vague sets are regarded as a special case of context-dependent fuzzy sets. Definition 2.2: [2] Let A and B be VSs of the form The basic concepts of vague set theory and its extensions defined by [3,6]. In 1970, Levine [7] initiated the study of A x,tA x,1 f A x / x X and generalized closed sets in topological spaces. The concept of B x,t x,1 f x / x X. Then fuzzy sets was introduced by Zadeh [12] in 1965. The theory B B of fuzzy topology was introduced by C.L.Chang [4] in 1967; several researches were conducted on the generalizations of a) A B if and only if tA x tB x and the notions of fuzzy sets and fuzzy topology. In 1986, the 1 f x1 f x for all x X . concept of intuitionistic fuzzy sets was introduced by A B Atanassov [1] as a generalization of fuzzy sets. b) A B if and only if and B A . c c) A x,f x,1 t x / x X . In this paper we introduce the concept of vague contra A A generalized pre continuous mapping, vague generalized pre d) A B x,tA x tB x,1 f A x 1 fB x / x X homeomorphism and vague M-generalized pre homeomorphism and we also obtain their properties and relations with counter examples. e) A B x,tA x tB x,1 f A x 1 fB x / x X II. PRELIMINARIES For the sake of simplicity, we shall use the notion Definition 2.1:[2] A vague set A in the universe of discourse X is characterized by two membership functions given by: A x,tA x,1 f A x instead of . i) A true membership function tA : X 0,1 and ii) A false membership function f A : X 0,1. Definition 2.3:[10] A vague topology (VT in short) on X is a family of vague sets (VS in short) in X satisfying the where t A x is lower bound of the grade of membership of following axioms. x derived from the “evidence for x ”, and f A x is a lower bound of the negation of x derived from the “evidence 0, 1 G G ,for any G ,G against ” and tA x f A x 1. Thus the grade of 1 2 1 2 membership of x in the vague set is bounded by a Gi foranyfamily Gi /i J. subinterval t x ,1 f x of [0,1]. This indicates that if A A In this case the pair X , is called a vague topological the actual grade of membership x, then space (VTS in short) and any VS in is known as a vague tA x x 1 f A x. The vague set is written as open set (VOS in short) in . A x,tA x,1 f A x / x X where the interval International Journal of Engineering Science and Computing, August 2016 2611 http://ijesc.org/ The complement c of a VOS in a VTS is called a mapping if 1 is a VGPCS in for every VGPCS A f A vague closed set (VCS in short) in . A in . Definition 2.4:[10] A VS A x,tA,1 f A in a VTS Definition 2.7:[11] Let X , be a VTS. The vague is said to be a generalized pre closure ( vgpclA in short) for any VS A is defined as follows, i) vague semi closed set (VSCS in short) if vintvclA A, vgpclA K / K is a VGPCSin X and A K. If ii) vague pre- closed set (VPCS in short) if is VGPCS, then vgpclA A . vclvintA A, iii) vague -closed set (V CS in short) if Definition 2.8:[10] A VTS X , is said to be a vague pT1/2 vclvintvclA A, space (V pT1/2 in short) if every VGPCS in is a VCS in iv) vague regular closed set (VRCS in short) if . A vclvintA. Definition 2.9:[10] A VTS is said to be a vague Definition 2.5:[11] Let X , and Y, be any two vague gpT1/2 space (V gpT1/2 in short) if every VGPCS in is a topological spaces. A map f : X, Y, is said to be VPCS in . 1 vague continuous (V continuous in short) if f V is Definition 2.10:[5] Let and be any two vague closed set in for every vague closed set V of topological spaces. A map is said to be . contra continuous if is closed set in for vague semi-continuous (VS continuous in short) if every open set in . is vague semi-closed set in for every Definition 2.11:[8] Let f be a bijective mapping from a vague closed set of . topological space X , into a topological space Y, . vague pre-continuous (VP continuous in short) if Then is said to be generalized homeomorphism if is vague pre-closed set in for every vague closed 1 f and f are generalized continuous mapping. set of . vague -continuous (V -continuous in short) if Definition 2.12:[9] A map f :X, Y, is said to be is vague -closed set in for every vague generalized pre closed if f V is generalized pre closed set closed set of . in for every closed set in . vague generalized continuous (VG continuous in short) if is vague generalized closed set in for III. VAGUE HOMEOMORPHISM IN TOPOLOGICAL every vague closed set of . SPACES vague generalized semi-continuous (VGS continuous in X Definition 3.1: Let be a bijective mapping from a VTS short) if is vague generalized semi-closed set in into a VTS . Then is said to be for every vague closed set of . vague -generalized continuous (V G continuous in i) Vague homeomorphism (V homeomorphis m in short) if short)if is vague -generalized closed set in are V continuous mapping. ii) Vague pre homeomorphism (VP homeomorphism in for every vague closed set of . X1 , short) if f and f are VP continuous mapping. iii) Vague generalized homeomorphism (VG Definition 2.6:[11] A map is said to be homeomorphism in short) if are VG vague generalized pre irresolute (VGP irresolute in short) continuous mapping. International Journal of Engineering Science and Computing, August 2016 2612 http://ijesc.org/ Definition 3.2: A map is said to be are VGP continuous mappings. That is the vague generalized pre closed if is vague generalized mapping is a VGP homeomorphism. pre closed set in for every vague closed set in . Example 4.6: Let and IV. VAGUE GENERALIZED PRE HOMEOMORPHISM G1 x,0.2,0.3,0.2,0.4 , . Then and Definition 4.1: A bijection mapping f :X, Y, is G2 x,0.3,0.5,0.1,0.3 called a vague generalized pre homeomorphism (VGP are VTs on and respectively. Define a homeomorphism in short) if are VGP continuous bijection mapping by mapping. Then is a VGP f a u and f b v. Example 4.2: Let X a,b, Y u,v and homeomorphism but not VP homeomorphism since pT1/2 are not a VP continuous mapping. G1 x,0.3,0.5,0.3,0.6 , G x, 0.3,0.4 , 0.2,0.6 . Then and 2 0,G1,1 Theorem 4.7: Let be a VGP 0,G ,1 are VTs on and Y respectively. Define a homeomorphism, then is a V homeomorphism if 2 X , Y, bijection mapping by f a u X andY are V space. f : X, Y , and f b v. Then f is a VGP continuous mapping and 1 1 Proof: Let B be a VCS in Y . Then f B is a VGPCS in is also a VGP continuous mapping. Therefore is a f VGP homeomorphism. V X , by hypothesis. Since X is a V space, is a VCS in . Hence is a V continuous mapping. By Theorem 4.3: Every V homeomorphism is a VGP homeomorphism but not conversely. hypothesis f 1 : Y, X, is a VGP continuous 1 1 Proof: Let be a V homeomorphism. mapping. Let A be a VCSf in X . Then f A f A Then are V continuous mappings. This implies is a VGPCS in , by hypothesis. Since is a V space, are VGP continuous mappings. That is the f A is a VCS in . Hence is a V continuous mapping is a VGP homeomorphism. mapping. Therefore the mapping is a V homeomorphism. Example 4.4: Let and Theorem 4.8: Let f :X, Y,be a VGP homeomorphism, then isf V a VP homeomorphis m if G1 x,0.6,0.7,0.5,0.8 , are V T space.
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