Set operations of fuzzy sets using gradual elements

Abstract

The conventional set operations of fuzzy sets are based on the membership functions using the $$\max $$ and $$\min $$ functions. In this paper, we shall consider the set operations of fuzzy sets based on the concepts of gradual sets and gradual elements. When the fuzzy sets can be formulated as consisting of gradual elements like the usual set consisting of usual elements, the intersection and union of fuzzy sets can be defined as the same way as the intersection and union of usual sets. In this case, the set operations of fuzzy sets will be similar to the set operations of crisp sets.