Functional degrees of inclusion and similarity between L-fuzzy sets

Abstract

Inclusion is one of the most basic relations between sets. In this paper, we show how to represent the degree of inclusion between two L-fuzzy sets via a function. Specifically, such a function determines the minimal modifications needed in an L-fuzzy set to be included (in Zadeh's sense) into another. To reach such a goal, firstly we present the notion of f-inclusion, which defines a family of crisp binary relations between L-fuzzy sets that are used as indexes of inclusion and, subsequently, we define the φ-degree of inclusion as the most suitable f-inclusion under certain criterion. In addition, we also present three φ-degrees of similarity definable from the φ-degree of inclusion. We show that the φ-degree of inclusion and the φ-degrees of similarities satisfy versions of many common axioms usually required for measures of inclusion and similarity in the literature.