Entropy of multiple sets

Abstract

Multiple set theory is a generalization of the existing set theoretical concepts like fuzzy set, multiset, fuzzy-multiset and multi-fuzzy set. The limitations of the crisp set theory in the real world applications and comparison of the set objects with respect to certain characteristics have led to the introduction of fuzzy set theory. As the name indicates, it discusses the fuzziness of the objects in the set. Fuzzy sets are characterised by the fuzzy membership functions with co-domain [0, 1]. With an order of (n, k), a multiple set helps to consider n distinct characteristics on the set simultaneously. In multiple sets each element is assigned with a membership matrix of order n × k. For different values of n and k multiple set becomes equivalent to fuzzy set, multiset, multi-fuzzy set and fuzzy-multi set. Multiple sets satisfy the basic set operations except the laws of contradiction and excluded middle. Multiple sets find its application in medical diagnosis, pattern recognition etc.The measure of fuzziness is termed as the entropy of a fuzzy set. An entropy is basically a function from the collection of all fuzzy sets to non negative real number fields which satisfies certain properties. Since fuzzy sets are generalized by multiple sets, the concept of fuzzy entropy entropy can be extended to multiple sets as well. The analog of fuzzy set entropy in multiple sets is called multiple set entropy. This paper concentrates on the properties and examples of multiple set entropy. It also discusses similarity measure and distance measure of fuzzy sets as well as multiple sets. This paper introduces the concept of interval entropy, which enables the value of entropy to be an interval rather than a number, and provides examples for the same.