Encyclopedia of Database Systems

2018 Edition
| Editors: Ling Liu, M. Tamer Özsu

Fuzzy Relation

  • Vilém Novák
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-8265-9_5010

Definition

An n-ary fuzzy relation R is a fuzzy set (see “ Fuzzy Set”) in the universe U1 × … × Un, i.e., RU1 × … × Un. A special case is a binary fuzzy relation in the universe U × V, i.e., RU × V.

Key Points

Mathematically, an n-ary fuzzy relation is a function R: U1 × … × UnL where L is a residuated lattice.

Let RU × V and SV × W be two binary fuzzy relations. The composition of R and S is a fuzzy relation RS determined by the membership function
$$ \begin{array}{c}\big(R\circ S\big)\left(x,z\right)=\underset{y\in V}{\vee}\left(R\left(x,y\right)\wedge S\left(y,z\right)\right),\hfill \\ {}\hfill \kern11.5em x\in U,z\in W.\hfill \end{array} $$
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Recommended Reading

  1. 1.
    Klement EP, Mesiar R, Pap E. Triangular norms. Dordrecht: Kluwer; 2000.zbMATHCrossRefGoogle Scholar
  2. 2.
    Klir GJ, Yuan B. Fuzzy sets and fuzzy logic: theory and applications. New York: Prentice-Hall; 1995.zbMATHGoogle Scholar
  3. 3.
    Novák V. Fuzzy sets and their applications. Bristol: Adam Hilger; 1989.zbMATHGoogle Scholar
  4. 4.
    Novák V, Perfilieva I, Močkoř J. Mathematical principles of fuzzy logic. Boston/Dordrecht: Kluwer; 1999.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Research and Applications of Fuzzy ModelingUniversity of OstravaOstravaCzech Republic