Crisp sets as classes of discontinuous fuzzy sets

Enric Trillas, Itziar García-Honrado, Sergio Guadarrama, Eloy Renedo

Abstract: This paper aims to show how, by using a threshold-based approach, a path from imprecise information to a crisp ‘decision’ can be developed. It deals with the problem of the logical transformation of a fuzzy set into a crisp set. Such threshold arises from the ideas of contradiction and separation, and allows us to prove that crisp sets can be structurally considered as classes of discontinuous fuzzy sets. It is also shown that continuous fuzzy sets are computationally indistinguishable from some kind of discontinuous fuzzy sets.