Identities and Inequalities on Fuzzy Set Cardinalities

In this paper, we study identities and inequalities on fuzzy set cardinalities. An important such identity is the valuation property. Other identities are related to the fuzzification of the (symmetric) difference of fuzzy sets. We show that using a stable, commutative quasi-copula and its dual operator for modelling intersection and union of fuzzy sets, identities which are valid in the crisp case, also hold for cardinalities of fuzzy sets. The basic inequalities that come to mind are the Bell inequalities. We show that our results on the fuzzified Bell inequalities can be exploited to develop a framework in which the validity of more general inequalities on fuzzy cardinalities can be checked easily. An interesting application of these meta-theorems can be found in the field of similarity measurement