Fuzzy Sets in Probabilistic Structural Mechanics

Abstract

Almost 30 years ago, Zadeh (1965) introduced the concept of fuzzy sets. In ordinary set theory, an element is either a member or not a member of a set (in logic, it is indicated by either 1 or 0). In the fuzzy set, whether or not an element belongs to a set may be expressed not just by 1 or 0, but by any value between 0 and 1, indicating different degrees of the membership of the element. For example, 0 indicates a nonmembership, 0.3 a weak membership, 0.9 a strong membership, and 1.0 crisp membership. Building on Zadeh’s work, Yager (1986) offers the following definition: A fuzzy set is a generalization of the ideas of an ordinary or crisp set. A fuzzy subset can be seen as a predicate whose truth values are drawn from the unit interval, I = [0, 1] rather than the set (0, 1) as in the case of an ordinary set. Thus fuzzy subset has as its underlying logic a multi-valued logic.