Definition of a fuzzy (sub)set

Definition of a Fuzzy (Sub)Set

A crisp subset A of $\Omega$ is defined by a characteristic function $\chi_a$ which takes a value 0 for the elements of $\Omega$ which do not belong to A and a value 1 for those which do belong to A:

$\displaystyle \chi_A(x)$ : $\displaystyle \Omega \longrightarrow \{0,1\}$

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$\displaystyle \chi_A(x)$ = $\displaystyle \left\{\begin{array}{l}0 \text{ if } x \notin A \\1 \text{ if } x \in A\end{array}\right.$

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A fuzzy subset A of $\Omega$ is defined by a membership function which associates to each element x of $\Omega$ , the $\mu_a(x)$ degree (between 0 and 1) to which x belongs to A:

$\begin{displaymath}\mu_A(x) : \Omega \longrightarrow [0,1]\end{displaymath}$

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In the particular case where $\mu_a$ takes only values equal to 0 or 1, the fuzzy subset A is a crisp subset of $\Omega$ . A crisp subset is thus a particular case of fuzzy subset.

To represent the fuzzy subset A, we often use a notation indicating, for any element x of $\Omega$ , its degree $\mu_a(x)$ of membership to A:

A = $\displaystyle \sum_{x \in \Omega} \frac{\mu_A(x)}{x} \quad \text{if } \Omega\text{ is finite,}$

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A = $\displaystyle \int \frac{\mu_A(x)}{x} dx \quad \text{if } \Omega\text{ is infinite.}$

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