Measures of inclusion and entropy based on the φ-index of inclusion

Surprisingly, despite that fuzzy sets were introduced more than fifty years ago, there is not consensus yet about how to extend the notion of inclusion in such a framework. Recently, alternatively to previous methods in the literature, we introduced an approach in which we make use of the so-called phi-index of inclusion. This approach has a main difference with respect to previous ones: the degree of inclusion is identified with a function instead of with a value in [0, 1], although such a feature makes it difficult to compare the phi-index of inclusion with existing axiomatic approaches concerning measures of inclusion. This is the reason why in this paper we define two different and natural measures of inclusion by means of the phi-index of inclusion and, then, show that both measures satisfy some standard axiomatic approaches about measures of inclusion in the literature. In addition, taking into account the relationship of fuzzy entropy with Young axioms for measures of inclusion, we present also a measure of entropy based on the phi-index of inclusion that is in accordance with the axioms of De Luca and Termini. (C) 2021 Elsevier B.V. All rights reserved.