The Concept of Fuzzy Relation and BasicProperties of its

This article introduces the fuzzy relation. This is the expansion of usual relation, and the value is given at the fuzzy value. At rst, the deenition of the fuzzy relation characterized by membership function is described. Next, the deenitions of the zero relation and universe relation and basic operations of these relations are shown. In this paper C 1 , C 2 are non empty sets. Let us consider C 1 , C 2. A partial function from : C 1 ; C 2 : ] to R is said to be a Membership function of C 1 , C 2 if: (Def. 1) dom it = : C 1 ; C 2 : ] and rng it 0; 1]: We now state the proposition (1) : C1; C2 : ]; : C1; C2 : ] is a Membership function of C 1 , C 2. Let C 1 , C 2 be non empty sets and let h be a Membership function of C 1 , C 2. A set is called a fuzzy relation of C 1 , C 2 , h if: is deened by: (Def. 3) For every element c of : C 1 ; C 2 : ] holds h(c) = g(c): Let C 1 , C 2 be non empty sets, let h, g be Membership functions of C 1 , C 2 , let A be a fuzzy relation of C 1 , C 2 , h, and let B be a fuzzy relation of C 1 , C 2 , g. The predicate A B is deened by: (Def. 4) For every element c of : C 1 ; C 2 : ] holds h(c) g(c): 1 c Association of Mizar Users