Some Considerations on Membership Functions of Fuzzy Sets

Abstract

In defining fuzzy sets, we focus on the values of membership but pay little attention to the values of the universe of discourse (variable). For example, consider the definition of the fuzzy set "large number" on real numbers. When we consider 50, we expect the membership values for 49, 50 and 51 to be clearly different near 200. We, however, expect the membership values for 199, 200 and 201 to differ only slightly for values distant from 50. Since it is often the case that the membership values of distant values are usually defined as 1 or 0, we take little notice of them and are not so concerned about them. However, the relationship between the universe of discourse and memberships differs between the neighborhood of the considered value and its distant values, and they change smoothly from one to the other. In this paper, we consider several definitions of the membership function to avoid these differences.