Similarity measure, entropy and distance measure of multiple sets

A multiple set is an extended version of a fuzzy set that simultaneously addresses an element’s multiplicity and uncertainty. In this paper, we define the approximate equality of multiple sets and study some relevant properties associated with it. We then apply the notion of approximation equality of multiple sets to solve a pattern recognition problem. A novel class of similarity measures involving implication operators is introduced and the characteristics of approximate equality corresponding to these similarity measures are discussed. Further, we propose the concepts of σ-entropy, σ-distance measure, and σ-similarity measure of multiple sets and illustrate these with some examples. Finally, we define the theory of similarity measures between elements in multiple sets.