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

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

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 $\mathrm{\neg }:L\to L$ 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.