Encyclopedia of Database Systems

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

Fuzzy Set

  • Vilém Novák
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-8265-9_5008

Definition

A fuzzy set is a function A: U → L where U is an ordinary set of elements called universe and L is a scale which is usually supposed to have the structure of a residuated lattice (see “ Residuated Lattice”). The function A is at the same time called also membership function, i.e., fuzzy set is identified with its membership function.

If xU is an element then A(x) ∈ L is a membership degree of x in the fuzzy set A. It can also be interpreted as a degree of truth of the fact that the element x belongs to the fuzzy set A. If U is a universe and A a fuzzy set then it is convenient to write AU.

Key Points

Fuzzy sets can be explicitly written in the form
$$ \left\{a/u|a\in L,u\in U\right\} $$
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Recommended Reading

  1. 1.
    Klir GJ, Yuan B. Fuzzy sets and fuzzy logic: theory and applications. New York: Prentice-Hall; 1995.zbMATHGoogle Scholar
  2. 2.
    Novák V. Fuzzy sets and their applications. Bristol: Adam Hilger; 1989.zbMATHGoogle Scholar
  3. 3.
    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