Basic Properties of Fuzzy Set Operation and Membership Function

For simplicity, we follow the rules: C denotes a non empty set, c denotes an element of C, f , h, g, h1 denote membership functions of C, A denotes a FuzzySet of C, f , B denotes a FuzzySet of C, g, D denotes a FuzzySet of C, h, D1 denotes a FuzzySet of C, h1, X denotes a Universal FuzzySet of C, and E denotes an Empty FuzzySet of C. We now state four propositions: (1) For every element x of C and for every membership function h of C holds 0 ¬ h(x) and h(x) ¬ 1. (2) For every element x of C holds (EMFC)(x) = 0 and (UMFC)(x) = 1. (3) For every c such that f(c) ¬ h(c) holds (max(f,min(g, h)))(c) = (min(max(f, g), h))(c). (4) If A ⊆ D, then A ∪ B ∩ D = (A ∪ B) ∩ D.