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A fuzzy set is a pair (,) where is a set (often required to be non-empty) and : [,] a membership function. The reference set (sometimes denoted by or ) is called universe of discourse, and for each , the value () is called the grade of membership of in (,).
Fuzzy mathematics is the branch of mathematics including fuzzy set theory and fuzzy logic that deals with partial inclusion of elements in a set on a spectrum, as opposed to simple binary "yes" or "no" (0 or 1) inclusion. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets. [1]
Fuzzy set operations are a generalization of crisp set operations for fuzzy sets. There is in fact more than one possible generalization. There is in fact more than one possible generalization. The most widely used operations are called standard fuzzy set operations ; they comprise: fuzzy complements , fuzzy intersections , and fuzzy unions .
The orange arrow (pointing at 0.2) may describe it as "slightly warm" and the blue arrow (pointing at 0.8) "fairly cold". Therefore, this temperature has 0.2 membership in the fuzzy set "warm" and 0.8 membership in the fuzzy set "cold". The degree of membership assigned for each fuzzy set is the result of fuzzification. Fuzzy logic temperature
Type-2 fuzzy sets and systems generalize standard Type-1 fuzzy sets and systems so that more uncertainty can be handled. From the beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word fuzzy, since that word has the connotation of much uncertainty.
Fuzzy classification is the process of grouping elements into fuzzy sets [1] whose membership functions are defined by the truth value of a fuzzy propositional function. [2] [3] [4] A fuzzy propositional function is analogous to [5] an expression containing one or more variables, such that when values are assigned to these variables, the expression becomes a fuzzy proposition.
Lotfi Aliasker Zadeh [5] (/ ˈ z ɑː d eɪ /; Azerbaijani: Lütfi Rəhim oğlu Ələsgərzadə; [6] Persian: لطفی علیعسکرزاده; [2] 4 February 1921 – 6 September 2017) [1] [3] was a mathematician, computer scientist, electrical engineer, artificial intelligence researcher, and professor [7] of computer science at the University of California, Berkeley.
The membership degree () quantifies the grade of membership of the element to the fuzzy set ~. The value 0 means that is not a member of the fuzzy set; the value 1 means that is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially.