Fuzzy subset features

Fuzzy Subset Features

To be able to describe a fuzzy subset A of $\Omega$ easily, only some of its features are used, basically those showing how much it differs from a crisp subset of $\Omega$ .

Support:

The first of these features is the support of A, i.e. all the elements of $\Omega$ which belong, at least a little, to A. It is noted support(A) and is the part of $\Omega$ on which the membership function of A is not null (figure 21):

$\begin{displaymath}support(A) = \left\{ x \in \Omega / \mu_A(x) \neq 0 \right\}\end{displaymath}$

(19)

Height:

The second feature of A is its height, noted h(A). It represents the highest degree with which an element of $\Omega$ belongs to A. It is the greatest value of its membership function:

$\begin{displaymath}h(A) = \sup_{x \in \Omega} \left( \mu_A(x) \right)\end{displaymath}$

(20)

Normalized fuzzy subsets represent an important family of fuzzy subsets, used in possibility theory. In this case, at least one element of $\Omega$ exists that belongs absolutely to $\Omega$ (with degree 1). More precisely, A is normalized if its height h(A) is 1.

Kernel:

All the elements which belong absolutely (with degree 1) to A constitute the kernel of A, noted kernel(A) (figure 21):

$\begin{displaymath}ker(A) = \left\{ x \in \Omega / \mu_A(x) = 1 \right\}\end{displaymath}$

(21)

If A is a crisp subset of $\Omega$ , it is normalized, and identical to its support and to its kernel.

Cardinality:

Another characteristic of fuzzy subset A of $\Omega$ , when $\Omega$ is finite, is its cardinality, which is the global degree with which elements of $\Omega$ belong to A. Cardinality is defined by:

$\begin{displaymath}\vert A\vert = \sum_{x \in \Omega} \mu_A(x)\end{displaymath}$

(22)

If A is a crisp subset of $\Omega$ , its cardinality represents the number of elements of A.

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