Metrics of symmetric difference on fuzzy sets based on R-implicators of the usual families of t-norms

Abstract

Symmetric differences of two fuzzy sets by means of R-implications operators of a norm have been studied and their cardinalities provided classes of dissimilarity measures on fuzzy sets. In this paper, we characterize t-norms which generate metrics from these measures. The obtained metrics are fuzzy versions of the well-known distance of cardinality of the symmetric difference of crisp sets. For an application point of view, we consider the seven usual parameterized families of t-norms and we use their R-implications to examine conditions on parameters under which these t-norms generate metrics on fuzzy sets from those measures.