About approximate inclusion and its axiomatization

This paper is devoted to an extension of the inclusion operator. The regular inclusion of A in B entails that any element of A is also a member of B. The idea suggested in this paper is to relax (or soften) the universal quantifier (all) into “almost all”. It is shown that diverse approaches to the relaxation can be envisaged in order to design an approximate inclusion whose result is either Boolean, or valued in the unit interval. A set of axioms that was previously proposed for the characterization of graded inclusion of fuzzy sets is revisited in order to take into account the specificities of approximate inclusion. A concrete usage of this type of inclusion is illustrated in the area of databases, with the approximate division of fuzzy relations.