Equality, inclusion of fuzzy subsets

Equality, Inclusion of Fuzzy Subsets

Two fuzzy subsets A and B (of a same set of reference $\Omega$ ) are equal if their membership functions take the same value at any point of $\Omega$ :

$\begin{displaymath}\forall x \in \Omega, \quad \mu_A(x) = \mu_B(x)\end{displaymath}$

(23)

$A \in \mathcal{F}(\Omega)$ (see Notation) is included in $B \in \mathcal{F}(\Omega)$ , denoted by $A \subseteq B$ , if any element x of $\Omega$ which belongs, even moderately, to A also belongs to B with a membership degree at least as great.

Their membership functions are such that:

$\begin{displaymath}\forall x \in \Omega, \quad \mu_A(x) \leqslant \mu_B(x)\end{displaymath}$

(24)

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