Talk:Fuzzy set
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Fuzzy sets are an extension of the classical set theory used in fuzzy logic.
They are intended to be, but they are not! Membership functions are just that - functions. Functions are defined in terms of sets. So the definition of a fuzzy set depends on (and therefore is not an extension of) the definition of a set. Blaise 21:53, 28 Apr 2005 (UTC)
According to Fuzzy set theory, classical sets are a form of fuzzy sets. This is not so otherway around. In a way the statment at the start of page is true. User:srinivasasha 18:07, 15 Apr 2005 (IST)
Blaise, (in the event that you ever read this) you may want to consider deleting your additions to several entries stating that fuzzy sets do not extend ordinary sets. First, axiomatizations of fuzzy sets which do not `depend on' ordinary set theory exist since 1967 (i.e. 2 years after Zadeh's definition and 38 years before you were here). Second, membership functions are a convenient model of fuzzy logic but they are not essential, as aptly shown in Petr Hajek's work. Third, even if one restricts oneself to the `popular' view of membership functions, your argument is a fallacious one. You say: Membership functions are functions, therefore special sets. My reply is: a set is something that satisfies the axioms of set theory, so it is something that cannot be considered isolated from the notions appearing in those axioms, like those of element, union, and so on. It is clear that fuzzy sets, *the way Fuzzy Set Theory regards them*, do not generally fulfil those axioms and so are generally not sets.
For example, in FST the union of two fuzzy sets is not the union of the graphs of their membership functions; the fact that unions of graphs of membership functions do fulfil the axioms of set theory is completely immaterial.
Another example, if A is a non-crisp fuzzy subset of a universe U, then, no matter how you define an `element', the axiom stating that two sets are equal when they have the same elements will fail (because then A would equal the ordinary set of its elements, which *is* crisp).155.210.232.88 16:47, 4 October 2006 (UTC)
