Intuitionistic Fuzzy Generalized Beta Continuous Mappings

In this paper we introduce intuitionistic fuzzy generalized beta continuous mappings and intuitionistic fuzzy generalized beta irresolute mappings. We investigate some of their properties. Also we provide some characterization of intuitionistic fuzzy generalized beta continuous mappings and intuitionistic fuzzy generalized beta irresolute mappings. INTRODUCTION Atanassov [1] introduced the notion of intuitionistic fuzzy sets. Using the notion of intuitionistic fuzzy sets, Coker [3] introduced the notion of intuitionistic fuzzy topological spaces. Intuitionistic fuzzy beta continuous mappings in intuitionistic fuzzy topological spaces are introduced by Coker[3]. In this paper we introduce intuitionistic fuzzy generalized beta continuous mappings and intuitionistic fuzzy generalized beta irresolute mappings and we provide some characterizations. Preliminaries Definition 2.1: [1] An intuitionistic fuzzy set (IFS in short) A in X is an object having the form A = {〈x, μA (x), νA(x)〉 / x∈ X} where the functions μA : X → [0,1] and νA: X → [0,1] denote the degree of membership (namely μA(x)) and the degree of non membership (namely νA(x)) of each element x ∈X to the set A, respectively, and 0 ≤ μA (x) + νA(x) ≤ 1 for each x ∈X. Denote by IFS (X), the set of all intuitionistic fuzzy sets in X. Definition 2.2: [1] Let A and B be IFSs of the form A = {〈x, μA (x), νA(x)〉 / x∈X} and B = { 〈x, μB (x), νB(x)〉 / x∈ X}. Then (a) A ⊆ B if and only if μA (x) ≤ μB (x) and νA(x) ≥ νB(x) for all x ∈X (b) A = B if and only if A ⊆ B and B ⊆ A (c) A = {〈 x, νA(x), μA(x)〉 / x ∈ X} (d) A ∩ B = {〈x, μA(x) ∧ μB(x), νA(x) ∨ νB(x)〉 / x ∈ X} (e) A ∪ B = {〈x, μA(x) ∨ μB(x), νA(x) ∧ νB(x)〉 / x ∈ X} For the sake of simplicity, we shall use the notation A = 〈x, μA, νA〉 instead of A = {〈x, μA(x), νA(x)〉 / x ∈ X}. The intuitionistic fuzzy sets 0~ = {〈x, 0, 1〉 / x ∈X} and 1~ = {〈x, 1, 0〉 / x ∈ X} are respectively the empty set and the whole set of X. Definition 2.3: [7] The IFS c(α , β ) = 〈 x, cα , c 1-β 〉 where α ∈ (0, 1] , β ∈ [ 0, 1) and α + β ≤ 1 is called an intuitionistic fuzzy point (IFP for short) in X. Definition 2.4: [3] An intuitionistic fuzzy topology (IFT for short) on X is a family τ of IFSs in X satisfying the following axioms. (i) 0~, 1~ ∈ τ (ii) G1 ∩ G2 ∈ τ for any G1, G2 ∈ τ (iii) ∪ Gi ∈ τ for any family {Gi / i ∈ J} ⊆ τ. In this case the pair (X, τ) is called an intuitionistic fuzzy topological space (IFTS in short) and any IFS in τ is known as an intuitionistic fuzzy open set (IFOS in short) in X. The complement A of an IFOS A in IFTS (X, τ) is called an intuitionistic fuzzy closed set (IFCS in short) in X. Definition 2.5:[3] Let (X, τ) be an IFTS and A = 〈x, μA, νA〉 be an IFS in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure are defined by int(A) = ∪ {G / G is an IFOS in X and G ⊆ A} cl(A) = ∩ {K / K is an IFCS in X and A ⊆ K} Note that for any IFS A in (X, τ), we have cl(A) = (int(A)) and int(A) = (cl(A)) [3].