Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 128

or, if only the support of μ is finite:

=
x ∈ Supp( μ )

| |

μ μ ( x ) .

This definition is consistent with the traditional concept of the cardinality of a
binary set. In the case of a fuzzy set, each point counts as an amount equal to its
membership level. The cardinality is also referred to as the power of the fuzzy set (for
example, [LUC 72]).

This definition can be extended to the case where S is not finite but is measur-

(such that able. Let M be a measure S dM ( x )=1). The cardinality of μ is then

S

defined by:

=
S

| μ | μ ( x ) dM ( x ) .

8.2.5. Fuzzy number

In this section, we will assume that S = R .

. A fuzzy interval is a convex fuzzy quantity
(all of its α -cuts are intervals). The upper semi-continuity of μ is equivalent to the fact
that the α -cuts are closed intervals.

A fuzzy quantity is a fuzzy set μ in R

A fuzzy number is an upper semi-continuous (u.s.c.) interval with a compact and
unimodal support. An example of a fuzzy number representing “roughly 10” is shown
in Figure 8.1.

μ

1

0

R

10

Figure 8.1. Fuzzy number representing “roughly 10”