Definition
An n-ary fuzzy relation R is a fuzzy set (see “Fuzzy Set”) in the universe U1 × … × Un, i.e., R ⫇ U1 × … × Un. A special case is a binary fuzzy relation in the universe U × V, i.e., R ⫇ U × V.
Key Points
Mathematically, an n-ary fuzzy relation is a function R: U1 × … × Un → L where L is a residuated lattice.
Let R ⫇ U × V and S ⫇ V × W be two binary fuzzy relations. The composition of R and S is a fuzzy relation R ∘ S determined by the membership function
The operation ∧ in (1) can be replaced by arbitrary t-norm.
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Recommended Reading
Klement EP, Mesiar R, Pap E. Triangular norms. Dordrecht: Kluwer; 2000.
Klir GJ, Yuan B. Fuzzy sets and fuzzy logic: theory and applications. New York: Prentice-Hall; 1995.
Novák V. Fuzzy sets and their applications. Bristol: Adam Hilger; 1989.
Novák V, Perfilieva I, Močkoř J. Mathematical principles of fuzzy logic. Boston/Dordrecht: Kluwer; 1999.
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Novák, V. (2018). Fuzzy Relation. In: Liu, L., Özsu, M.T. (eds) Encyclopedia of Database Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8265-9_5010
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