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 $s:S\rightarrow[0,1]$ 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 $\aleph_{1}^{|S|}$ fuzzy subsets of $S$.

The join and meet operations  in the complete lattice $[0,1]^{S}$ are named union and , respectively. The operation induced by the involution is called . 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

 $(s\cup t)(x)=\max\{s(x),t(x)\}\,\,\,;\,\,\,(s\cap t)(x)=\min\{s(x),t(x)\}\,\,% \,;\,\,\,-s(x)=1-s(x).$

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 $\neg:L\rightarrow L$ is defined in $L$, the complement $-s$ of $s$ is defined by setting

 $-s(x)=\neg s(x).$

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., . (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
• 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) 338ÃÂ­-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 FuzzySubset 2013-03-22 16:34:54 2013-03-22 16:34:54 ggerla (15808) ggerla (15808) 11 ggerla (15808) Definition msc 03E72 msc 03G20 fuzzy set L-subset Logic FuzzyLogic2