The Concept of Fuzzy Set and Membership Function and Basic Properties of Fuzzy Set Operation

This article introduces the fuzzy theory. At first, the definition of fuzzy set characterized by membership function is described. Next, definitions of empty fuzzy set and universal fuzzy set and basic operations of these fuzzy sets are shown. At last, exclusive sum and absolute difference which are special operation are introduced. [6] provide the notation and terminology for this paper. In this paper x, y are sets, C is a non empty set, and c is an element of C. The following proposition is true (1) rng(χ x,y) ⊆ [0, 1]. Let us consider C. A partial function from C to R is said to be a membership function of C if: (Def. 1) dom it = C and rng it ⊆ [0, 1]. Next we state the proposition (2) χ C,C is a membership function of C. In the sequel f , h, g, h 1 denote membership functions of C. Let C be a non empty set and let h be a membership function of C. A set is called a FuzzySet of C, h if: (Def. 2) It = [:C, h • C :]. Let C be a non empty set, let h, g be membership functions of C, let A be a FuzzySet of C, h, and let B be a FuzzySet of C, g. The predicate A = B is defined as follows: (Def. 3) h = g. Let C be a non empty set, let h, g be membership functions of C, let A be a FuzzySet of C, h, and let B be a FuzzySet of C, g. The predicate A ⊆ B is defined by: (Def. 4) For every element c of C holds h(c) ≤ g(c).