Fuzzy Sets

Fuzzy sets were introduced by Lofti Zadeh as a generalization of the concept of a regular set. A fuzzy set is characterized by a membership function that assigns a degree (or grade) of membership to all the elements in the universe of discourse. The membership value is a real number in the range [0, 1], where 0 denotes no definite membership, 1 denotes definite membership, and intermediate values denote partial membership to the set. In this way, the transition from nonmembership to membership in a fuzzy set is gradual and not abrupt like in a regular set, allowing the representation of imprecise concepts like “small,” “cold,” “large,” or “very” for example.

A variable with its values defined by fuzzy sets is called a linguistic variable. For example, a linguistic variable used to represent a temperature can be defined as taking the values “cold,” “comfortable,” and “warm,” each one of them defined as a fuzzy set. These linguistic labels, which are imprecise by their own nature, are,...