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Gradual elements in a fuzzy set. (English) Zbl 1133.03026

A new concept in fuzzy set theory is introduced – gradual element. Motivations are given from pure and applied mathematics supporting the necessity to introduce this concept. The relationship between gradual elements and elements is clarified – a gradual element is to an element of a set what a fuzzy set is to a set; a gradual element is a “flexible” element (depending on the relevance value), while an element is fixed. It is shown that connectives of fuzzy set theory are consistent with the notions of gradual element and gradual set. The potential of the notion of gradual element is sketched on some ill-understood aspects of fuzzy set theory and its applications, like fuzzy cardinality, fuzzy optimization, algebraic structures, fuzzy intervals, defuzzification. The new concept fills a gap in fuzzy set theory.

MSC:

03E72 Theory of fuzzy sets, etc.
68T37 Reasoning under uncertainty in the context of artificial intelligence
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