Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 125

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 ,μ ( 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 ,μ ( 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 ,μ ( 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 ,μ ( 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 ,μ ( x ) ≤ ν ( x ) . [8.6]

The equality of μ and ν is of course equivalent to having the inclusion in both
directions.