Digital Signal Processing Reference
In-Depth Information
X
=
{
x
|
x>
8
}
(7.1)
where 8 represents an unambiguous boundary. On the other hand, a
fuzzy set does not have a crisp boundary. To represent this fact, a new
concept is introduced, that of a membership function describing the
smooth transition from the fact “belongs to a set” to “does not belong
to a set”. Fuzzyness stems not from the randomness of the members of
the set but from the uncertain nature of concepts.
This chapter will review some of the basic notions and results in
fuzzy set theory.
Fuzzy systems are described by fuzzy sets and operations on fuzzy
sets. Fuzzy logic approximates human reasoning by using linguistic
variables and introduces rules based on combinations of fuzzy sets by
these operations. The notion of fuzzy set way introduced by Zadeh [295].
Crisp sets
Definition 7.1:
Crisp set
Let
X
be a non empty set considered to be the
universe of discourse
.
A
crisp set A
is defined by enumerating all elements
x
∈
X
,
A
=
{
x
1
,x
2
,
···
,x
n
}
(7.2)
that belong to
A
.
The universe of discourse consists of ordered or nonordered discrete
objects or of the continuous space.
Definition 7.2:
Membership function
The
membership function
can be expressed by a function
u
A
,that
maps
X
on a binary value described by the set
I
=
{
0
,
1
}
:
u
A
(
x
)=
1if
x
∈
A
u
A
:
X
→
I,
(7.3)
0 f
x
∈
A.
Here,
u
A
(
x
)representsthe
membership degree
of
x
to
A
.
Thus, an arbitrary
x
either belongs to
A
or it does not; partial member-
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