An Etymological Dictionary of Astronomy and Astrophysics

A set of → truth values in → fuzzy logic that does not have sharp boundaries. Instead, set members have degrees of membership. If the value of 1 is assigned to objects entirely within the set and a 0 is assigned to objects outside of the set, then any object partially in the set will have a value between 0 and 1. This contrast with → crisp sets in → classical logic where members assume a precise value of 1 or 0. Fuzzy sets were first introduced by Lotfi A. Zadeh (1965) and defined as follows. Let X be a space of points, with a generic element of X denoted by x. Thus X = {x}. A fuzzy set A in X is characterized by a → membership function  f_A(x) which associates with each point in X a real number in the interval [0,1], with the values of f_A(x) at x representing the “grade of membership” of x in A. Thus, the nearer the value of f_A(x) to unity, the higher the grade of membership of x in A. Generally, the intersection operations of fuzzy sets are the expansion of that operation on → nonfuzzy sets. In other words, operations on nonfuzzy sets are a particular case of operations on fuzzy sets.

See also: → fuzzy; → set.