On rational cardinality-based inclusion measures

Authors:
B. De Baets;H. De Meyer;H. Naessens
Affiliations:
Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, B-9000 Gent, Belgium;Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000 Gent, Belgium;Hogeschool Gent, Schoonmeersstraat 52, B-9000 Gent, Belgium
Venue:
Fuzzy Sets and Systems
Year:
2002

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Abstract

In order to express the degree to which a subset of a finite universe is contained into another subset, the concept of inclusion measure (or subsethood measure) of ordinary sets is introduced. A distinction is made between three types of inclusion measures. The first type yields reflexive inclusion measures, whereas the second and third type both give rise to locally reflexive inclusion measures, the latter ones simply being complementary to the former ones.Furthermore, a systematic way of generating inclusion measures for ordinary sets is presented in the form of a rational expression solely based on cardinalities of the sets involved. Various properties of the obtained rational inclusion measures, such as monotonicity and transitivity, are investigated.