Fuzzy sets and systems: theory and applications

Fuzziness is not a priori an obvious concept and demands some explanation. “Fuzziness” is what Black (NF 1937) calls “vagueness” when he distinguishes it from “generality” and from “ambiguity.” Generalizing refers to the application of a symbol to a multiplicity of objects in the field of reference, ambiguity to the association of a finite number of alternative meanings having the same phonetic form. But, the fuzziness of a symbol lies in the lack of well-defined boundaries of the set of objects to which this symbol applies. More specifically, let X be a field of reference, also called a universe of discourse or universe for short, covering a definite range of objects. Consider a subset à where transition between membership and nonmembership is gradual rather than abrupt. This “fuzzy subset” obviously has no well-defined boundaries. Fuzzy classes of objects are often encountered in real life. For instance, à may be the set of tall men in a community X. Usually, there are members of X who are definitely tall, others who are definitely not tall, but there exist also borderline cases. Traditionally, the grade of membership 1 is assigned to the objects that completely belong to ×here the men who are definitely tall; conversely the objects that do not belong to à at all are assigned a membership value 0. Quite naturally, the grades of membership of the borderline cases lie between 0 and 1. The