Generalized representation theorem and its application to the construction of fuzzy sets: Existence and uniqueness

A new methodology for constructing a fuzzy subset of a universal set U from a family of subsets of U based on a pre-determined range S that is a subset of (0,1] is proposed in this paper. The uniqueness of this construction method is also investigated, which depends on the structure of subset S. Whether the supremum of S lies in S is an important issue. This concern will be thoroughly studied in this paper.