Fuzzy set-theoretic operators and quantifiers

Janos Fodor

Ronald R. Yager

pp. 125-193

in: Didier Dubois, Henri Prade (eds), Fundamentals of fuzzy sets, Berlin, Springer, 2000

Abstract

This chapter summarizes main ways to extend classical set-theoretic operations (complementation, intersection, union, set-difference) and related concepts (inclusion, quantifiers) for fuzzy sets. Since these extensions are mainly pointwisely defined, we review basic results on the underlying unary or binary operations on the unit interval such as negations, t-norms, t-conorms, implications, coimplications and equivalences. Some strongly related connectives (means, OWA, weighted, and prioritized operations) are also considered, emphasizing the essential differences between these and the formerly investigated operator classes. We also show other operations which have no counterpart in the classical theory but play some important role in fuzzy sets (like symmetric sums, weak t-norms and conorms, compensatory AND).