fuzzy subset
Fuzzy set theory is based on the idea that vague notions as “big”, “near”, “hold” can be modelled by “fuzzy subsets”. The idea of a fuzzy subset $T$ of a set $S$ is the following: each element $x\in S$, there is a number $p\in [0,1]$ such that ${p}_{x}$ is the “probability” that $x$ is in $T$.
To formally define a fuzzy set, let us first recall a well-known fact about subsets: a subset $T$ of a set $S$ corresponds uniquely to the characteristic function^{} ${c}_{T}:S\to \{0,1\}$, such that ${c}_{T}(x)=1$ iff $x\in T$. So if one were to replace $\{0,1\}$ with the the closed unit interval $[0,1]$, one obtains a fuzzy subset:
A fuzzy subset of a set $S$ is a map $s:S\to [0,1]$ from $S$ into the interval $[0,1]$.
More precisely, the interval $[0,1]$ is considered as a complete lattice^{} with an involution $1-x$. We call fuzzy subset of $S$ any element of the direct power ${[0,1]}^{S}$. Whereas there are ${2}^{|S|}$ subsets of $S$, there are ${\mathrm{\aleph}}_{1}^{|S|}$ fuzzy subsets of $S$.
The join and meet operations^{} in the complete lattice ${[0,1]}^{S}$ are named union and intersection^{}, respectively. The operation induced by the involution is called complement^{}. This means that if $s$ and $t$ are two fuzzy subsets, then the fuzzy subsets $s\cup t,s\cap t,-s$, are defined by the equations
$$(s\cup t)(x)=\mathrm{max}\{s(x),t(x)\};(s\cap t)(x)=\mathrm{min}\{s(x),t(x)\};-s(x)=1-s(x).$$ |
It is also possible to consider any lattice^{} $L$ instead of $[0,1]$. In such a case we call $L$-subset of $S$ any element of the direct power ${L}^{S}$ and the union and the intersection are defined by setting
$$(s\cup t)(x)=s(x)\vee t(x);(s\cap t)(x)=s(x)\wedge t(x)$$ |
where $\vee $ and $\wedge $ denote the join and the meet operations in $L$, respectively. In the case an order reversing function $\mathrm{\neg}:L\to L$ is defined in $L$, the complement $-s$ of $s$ is defined by setting
$$-s(x)=\mathrm{\neg}s(x).$$ |
Fuzzy set theory is devoted mainly to applications. The main success is perhaps fuzzy control.
References
- 1 Cignoli R., D Ottaviano I. M. L. and Mundici D.,Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht, (1999).
- 2 Elkan C., The Paradoxical Success of Fuzzy Logic^{}. (November 1993). Available from http://www.cse.ucsd.edu/users/elkan/http://www.cse.ucsd.edu/users/elkan/ Elkan’s home page.
- 3 Gerla G., Fuzzy logic: Mathematical tools for approximate reasoning, Kluwer Academic Publishers, Dordrecht, (2001).
- 4 Goguen J., The logic of inexact concepts^{}, Synthese, vol. 19 (1968/69)
- 5 Gottwald S., A treatise on many-valued logics, Research Studies Press, Baldock (2000).
- 6 HÃÂ¡jek P., Metamathematics^{} of fuzzy logic. Kluwer (1998).
- 7 Klir G. , UTE H. St.Clair and Bo Yuan, Fuzzy Set Theory Foundations and Applications, (1997).
- 8 Zimmermann H., Fuzzy Set Theory and its Applications (2001), ISBN 0-7923-7435-5.
- 9 Zadeh L.A., Fuzzy Sets, Information and Control, 8 (1965) 338ÃÂ-353.
- 10 Zadeh L. A., The concept of a linguistic variable and its application to approximate reasoning I, II, III, Information Sciences, vol. 8, 9(1975), pp. 199-275, 301-357, 43-80.
Title | fuzzy subset |
---|---|
Canonical name | FuzzySubset |
Date of creation | 2013-03-22 16:34:54 |
Last modified on | 2013-03-22 16:34:54 |
Owner | ggerla (15808) |
Last modified by | ggerla (15808) |
Numerical id | 11 |
Author | ggerla (15808) |
Entry type | Definition |
Classification | msc 03E72 |
Classification | msc 03G20 |
Synonym | fuzzy set |
Synonym | L-subset |
Related topic | Logic |
Related topic | FuzzyLogic2 |