Identification of fuzzy sets with a class of canonically induced random sets

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Any random subset of a space X clearly determines the membership function of a fuzzy subset through its one point coverages. This paper shows, conversely, that any fuzzy subset A of X can always be identified with, in general, many random subsets S(A) of X with respect to one point coverages i.e., simultaneously, for all x ϵ X, ϕA(x) = Pr(xϵS(A)). In a related manner, it is shown that any fuzzy set can be uniformly closely approximated with respect to one point coverages by a random set having a finite number of outcomes. In particular, the canonical mapping SU, defined by A → SU (A) =df ϕA <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sub> ([U,1]), where A is any fuzzy subset of X and U is any uniformly distributed r.v. over [0, 1], produces such an identification. Moreover, SU is an isomorphism from the collection of all fuzzy subsets onto a proper subcollection of all random subsets of X, with respect to many of the basic fuzzy set operations and corresponding ordinary set operations among random sets. In addition, SU, among all possible mappings from the class of all fuzzy subsets of X into the class of all random subsets of X which preserve one point coverages, induces both the maximal lower probability measure and the minimal upper probability measure in Dempster's sense on P (X). Applications of the results to fuzzy attribute reasoning are presented, emphasizing the close connection between fuzzy and random confidence sets.