Continuous T-norms and T-conorms Satisfying the Principle of Inclusion and Exclusion

Abstract

The classical principle of inclusion and exclusion is formulated for set-theoretic union and intersection. It is natural to ask if it can be extended to fuzzy sets. The answer depends on the choice of fuzzy logical operations (which belong to the larger class of aggregation operators). Further, the principle can be generalized to interval-valued fuzzy sets, resp. IF-sets (Atanassov’s intuitionistic fuzzy sets). The principle of inclusion and exclusion uses cardinality of sets (which has a natural extension to fuzzy sets, interval-valued fuzzy sets and IF sets) or, more generally, a measure, which can be defined in different ways. We also point up the question of the domain of the measure which has been neglected so far.