On the properties of subsethood measures.

This paper explores the properties of subsethood measures, including boundary conditions, monotonicity, duality, and additivity. These properties can be used as candidates of axioms to define specific subsethood measures. The standard set theory introduces the set-inclusion relation with a qualitative nature, that is, a set is either a subset of the other or not. As a quantitative generalization, a subsethood measure considers the degree of inclusion. It should keep essential characteristics of the qualitative set-inclusion and satisfy generalized properties with the qualitative set-inclusion relation as a special case. A systematic study of the relationships between the qualitative and quantitative frameworks results in the four types of properties of subsethood measures presented in this paper. A combination of different types of properties helps in constructing an appropriate subsethood measure in real-world applications.