The structure analysis of fuzzy sets

Since the naissance of the fuzzy set theory in 1965, it has been applied to many areas extensively. In the applications, people are confronted with the interpretation problem of membership functions frequently. Since each person has his/her own opinion about the meaning of a subjective concept, he/she always has his/her own membership function for the same concept. The applications of the theory, for example fuzzy control and fuzzy reasoning, show the robustness of the (more or less) optionally chosen membership functions. This phenomenon probably reflects the inherent characteristics of fuzzy sets. In order to uncover the reason, many researchers have pay attention to the interpretation of membership functions (fuzzy sets). In this paper, from the quotient space theory and fuzzy equivalence relation, a new structural definition of fuzzy set is given. The “isomorphic principle” and “similarity principle” of fuzzy sets, the necessary and sufficient conditions of the “isomorphism” and “ε-similarity” of two fuzzy equivalence relations, and some properties of the new structural definition are discussed. These results may open up some inherent properties of fuzzy sets and provide a new interpretation of the membership functions.