A class of rational cardinality-based similarity measures

A systematic way of generating similarity measures for ordinary sets is presented in the form of a rational expression solely based on cardinalities of the sets involved. Twenty-eight measures are examined carefully and completely classified on the basis of their boundary behaviour and properties of reflexivity and monotonicity. Two types of reflexivity (reflexivity and local reflexivity) and three types of monotonicity (involving, respectively, two, three and four sets) are considered. In addition, 17 of these measures are shown to be T-transitive, with the t-norm T ranging from the drastic product Z to the minimum operator M. The given class of rational cardinality-based measures covers some well-known similarity measures.