Fuzzification of crisp functions using gradual elements and gradual sets

Abstract

There are two ways to fuzzify the crisp functions into fuzzy functions. One is using the extension principle. Another one is using the expression in decomposition theorem. In this paper, we present a new methodology to fuzzify the crisp functions for non-normal fuzzy sets using the concepts of gradual elements and gradual sets. The main purpose of this paper is to establish the equivalence between the fuzzy functions that are obtained using the extension principle and the gradual elements when the membership functions are assumed to be upper semi-continuous. This kind of technique is based on the idea in which the fuzzy set is formulated as consisting of gradual elements like the usual set consisting of usual elements. In this case, the operations among sets can be borrowed to set up the operations among fuzzy sets.