In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh 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 are special cases of L-relations when L is the unit interval [ 0 , 1 ] {\displaystyle [0,1]} . They are now used throughout fuzzy mathematics, having applications in areas such as linguistics (De Cock, Bodenhofer & Kerre 2000), decision-making (Kuzmin 1982), and clustering (Bezdek 1978). 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 [ 0 , 1 ] {\displaystyle [0,1]} . 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 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 bioinformatics.
Definition A fuzzy set is a pair ( U , m ) {\displaystyle (U,m)} where U {\displaystyle U} is a set (often required to be non-empty) and m : U → [ 0 , 1 ] {\displaystyle m\colon U\rightarrow [0,1]} 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 x\in 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 x\in U} . Then x {\displaystyle x} is called
not included in the fuzzy set ( U , m ) {\displaystyle (U,m)} if m ( x ) = 0 {\displaystyle m(x)=0} (no member), fully included if m ( x ) = 1 {\displaystyle m(x)=1} (full member), partially included if 0 < m ( x ) < 1 {\displaystyle 0<m(x)<1} (fuzzy member). 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)} ).
Crisp sets related to a fuzzy set For any fuzzy set A = ( U , m ) {\displaystyle A=(U,m)} and α ∈ [ 0 , 1 ] {\displaystyle \alpha \in [0,1]} the following crisp sets are defined:
A ≥ α = A α = { x ∈ U ∣ m ( x ) ≥ α } {\displaystyle A^{\geq \alpha }=A_{\alpha }=\{x\in U\mid m(x)\geq \alpha \}} is called its α-cut (aka α-level set)
A > α = A α ′ = { x ∈ U ∣ m ( x ) > α } {\displaystyle A^{>\alpha }=A'_{\alpha }=\{x\in U\mid m(x)>\alpha \}} is called its strong α-cut (aka strong α-level set)
… excerpt ends here. Continue reading the full article.
