Definition of -cuts

Definition of $\alpha$ -Cuts

It is often interesting to refer to crisp subsets roughly corresponding to a given fuzzy subset (to use criteria of decision-making, for example).

The simplest way to perform this approximation is to fix a lower limit, noted $\alpha$ , with degrees of membership taken into account. Thus, the crisp subset $A_\alpha$ of $\Omega$ associated with A for threshold $\alpha$ is built by selecting all the elements of $\Omega$ which belong to A with a degree at least equal to $\alpha$ .

More precisely, for any value $\alpha$ of [0, 1], the $\alpha$ -cut $A_\alpha$ (or subset of level $\alpha$ ) of a fuzzy subset A of $\Omega$ is defined as subset $A_\alpha = \left\{ x \in \Omega/ \mu_A(x) \geqslant \alpha \right\}$ .

The associated characteristic function $\chi_{a_\alpha}$ is defined by:

$\begin{displaymath}\chi_{A_\alpha}(x) = 1 \text{ if and only if } \mu_A(x) \geqslant \alpha\end{displaymath}$

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