Complement of fuzzy subset

Complement of a Fuzzy Subset

The complement of an element A of $\mathcal{F}(\Omega)$ is defined by considering that an element x of $\Omega$ belongs all the more to $\complement_a$ that it belongs less to A (figure 22(d)).

It is defined like the fuzzy subset of $\Omega$ having the following membership function:

$\begin{displaymath}\forall X \in \Omega, \quad \mu_{\complement_A}(x) = 1 - \mu_A(x)\end{displaymath}$

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Unlike crisp subsets, we also have $\complement_a \cap A \neq \varnothing$ and $\complement_a \cup A \neq \Omega$ .

The other properties of classical set theory are however satisfied:

$\complement_{(A \cap B)} = \complement_A \cup \complement_B$ ;
$\complement_{(A \cup B)} = \complement_A \cap \complement_B$ ;
$\complement_{\left( \complement_A \right)} = A$

Examples

With the fuzzy subsets of figure 20, we get:

$\begin{displaymath}\complement_{A_1}= 0.2/C+ 0.4/L+ 0.6/U\end{displaymath}$

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