Fuzzy set

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, fuzzy sets (also known as uncertain sets) are <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> whose <a href="/facts/Element_(mathematics)/Q2mx1vHL">elements</a> have degrees of membership. Fuzzy sets were introduced independently by <a href="/facts/Lotfi_Asker_Zadeh/WojTlG4Y">Lotfi A. Zadeh</a> in 1965 as an extension of the classical notion of set.
At the same time, Salii (1965) defined a more general kind of structure called an "L-relation", which he studied in an <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebraic</a> context; 
fuzzy relations are special cases of L-relations when L is the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> [0, 1].
They are now used throughout <a href="/facts/Fuzzy_mathematics/DAdIA1HM">fuzzy mathematics</a>, having applications in areas such as <a href="/facts/Linguistics/oQuamvf2">linguistics</a> (De Cock, Bodenhofer & Kerre 2000), <a href="/facts/Decision_making/NbJYprSw">decision-making</a> (Kuzmin 1982), and <a href="/facts/Cluster_analysis/HUeDnWPt">clustering</a> (Bezdek 1978).
In classical <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, the membership of elements in a set is assessed in binary terms according to a <a href="/facts/Principle_of_bivalence/dnkYVCod">bivalent condition</a>—an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a <a href="/facts/Membership_function_(mathematics)/NlIzCXrR">membership function</a> valued in the <a href="/facts/Real_number/R02gw5Pb">real</a> unit interval [0, 1]. Fuzzy sets generalize classical sets, since the <a href="/facts/Indicator_function/QEuh04NM">indicator functions</a> (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only takes values 0 or 1. In fuzzy set theory, classical bivalent sets are usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as <a href="/facts/Bioinformatics/D5x2L8ee">bioinformatics</a>.

Fuzzy set open-in-new

Fuzzy set