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Fuzzy set operations based on the theory of falling shadows. (English) Zbl 0809.04008
In this contribution the authors provide a solid mathematical theory to support the well-known popular choices for the set-theoretic extension of union and intersection to fuzzy sets such as the max-min, the probabilistic sum-product, and the bounded sum-bold intersection combination. The construction is based upon the theory of falling shadows that has been developed by one of the authors [{\it Wang Peizhuang}: Fuzzy sets and falling shadows of random sets (Chinese) (1985; Zbl 0589.94014)]. The interpretation of the concepts of “cloud” and “falling shadow” is not quite clear. Moreover it is not shown if the relation between probability theory and fuzzy set theory is just a formal one. Finally we want to point out that the transition from $(\Omega, {\cal A}, P)$ to $([0,1], {\cal B}, m)$ is done without any motivation or explanation. The idea to interpret a fuzzy set in terms of probability theory is promising, but the way it is executed in this paper is very poor and it leaves a lot of unanswered questions. The given proofs are almost trivial while the terminology used is not very precise.
Reviewer: E.Kerre (Gent)

03E72Fuzzy set theory
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