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Fuzzy set

In mathematics, fuzzy sets (aka uncertain sets) are somewhat like sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh and Dieter Klaua 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 abstract algebraic context. Fuzzy relations, which are used now in different areas, such as linguistics (De Cock, Bodenhofer & Kerre 2000), decision-making (Kuzmin 1982), and clustering (Bezdek 1978), are special cases of L-relations when L is the unit interval .Bezdek, J.C. (1978). 'Fuzzy partitions and relations and axiomatic basis for clustering'. Fuzzy Sets and Systems. 1 (2): 111–127. doi:10.1016/0165-0114(78)90012-X..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:''''''''''''}.mw-parser-output .citation .cs1-lock-free a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em} In mathematics, fuzzy sets (aka uncertain sets) are somewhat like sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh and Dieter Klaua 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 abstract algebraic context. Fuzzy relations, which are used now in different areas, such as linguistics (De Cock, Bodenhofer & Kerre 2000), decision-making (Kuzmin 1982), and clustering (Bezdek 1978), are special cases of L-relations when L is the unit interval . In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — 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 membership function valued in the real unit interval . Fuzzy sets generalize classical sets, since the indicator functions (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take 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 bioinformatics. A fuzzy set is a pair ( U , m ) {displaystyle (U,m)} where U {displaystyle U} is a set and m : U → [ 0 , 1 ] {displaystyle mcolon U ightarrow } a membership function. The reference set U {displaystyle U} (sometimes denoted by Ω {displaystyle Omega } or X {displaystyle X} ) is called universe of discourse, and for each x ∈ U , {displaystyle xin U,} the value m ( x ) {displaystyle m(x)} is called the grade of membership of x {displaystyle x} in ( U , m ) {displaystyle (U,m)} . The function m = μ ( A ) {displaystyle m=mu (A)} is called the membership function of the fuzzy set A = ( U , m ) {displaystyle A=(U,m)} . For a finite set U = { x 1 , … , x n } , {displaystyle U={x_{1},dots ,x_{n}},} the fuzzy set ( U , m ) {displaystyle (U,m)} is often denoted by { m ( x 1 ) / x 1 , … , m ( x n ) / x n } . {displaystyle {m(x_{1})/x_{1},dots ,m(x_{n})/x_{n}}.} Let x ∈ U . {displaystyle xin U.} Then x {displaystyle x} is called The (crisp) set of all fuzzy sets on a universe U {displaystyle U} is denoted with S F ( U ) {displaystyle SF(U)} (or sometimes just F ( U ) {displaystyle F(U)} ). For any fuzzy set A = ( U , m ) {displaystyle A=(U,m)} and α ∈ [ 0 , 1 ] {displaystyle alpha in } the following crisp sets are defined:

[ "Fuzzy logic", "fuzzy image processing", "Fuzzy markup language", "max min composition", "linguistic summarization", "fuzzy index" ]
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