Image Processing Reference
In-Depth Information
The support of a fuzzy set μ is the set of points with strictly positive membership
to μ (it is a binary set):
Supp( μ )= x
( x ) > 0 .
∈S
[8.2]
The core of a fuzzy set μ is the set of points that are completely included in μ (it
is also a binary set):
Core( μ )= x
( x )=1 .
∈S
[8.3]
A fuzzy set μ is said to be normalized if at least one point is completely included
in the set μ (which is equivalent to Core( μ )
=
):
x
∈S
( x )=1 .
[8.4]
A fuzzy set μ is described as unimodal if there is only one point x such that
μ ( x )=1. A less restrictive definition allows the core to be a compact set, not neces-
sarily reduced to a point.
8.2.2. Set operations: Zadeh's original definitions
Since fuzzy sets were introduced to generalize the concept of sets, the first oper-
ations defined were set operations. In this section, we will present Zadeh's original
definitions [ZAD 65]. More general classes of operations will be presented in section
8.5.
The equality of two fuzzy sets is defined by the equality of their membership func-
tions:
μ = ν
⇐⇒ ∀
x
∈S
( x )= ν ( x ) .
[8.5]
The inclusion of a fuzzy set in another is defined by an inequality between the two
membership functions:
μ
ν
⇐⇒ ∀
x
∈S
( x )
ν ( x ) .
[8.6]
The equality of μ and ν is of course equivalent to having the inclusion in both
directions.
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