Digital Signal Processing Reference
In-Depth Information
A(x)
1
young
middle
aged
old
25 60
years
Figure 7.1
A membership function of temperature.
ship is not allowed.
For two sets
A
and
B
, combinations can be defined by the following
operations:
A
∪
B
=
{
x
|
x
∈
A
or
x
∈
B
}
(7.4)
A
∩
B
=
{
x
|
x
∈
A
and
x
∈
B
}
(7.5)
A
=
{
x
|
x
∈
A, x
∈
X
}
.
(7.6)
Additionally, the following rules have to be satisfied:
A
=
A
=
X
A
∪
∅
,
and
A
∩
(7.7)
Fuzzy sets
Definition 7.3:
Fuzzy set
Let
X
be a non-empty set considered to be the universe of discourse.
A
fuzzy set
is a pair (
X, A
), where
u
A
:
X
→
I
and
I
=[0
,
1].
Figure 7.1 is an example of a possible membership function.
The family of all fuzzy sets on the universe
x
will be denoted by
L
(
X
). Thus
L
(
X
)=
{
u
A
|
u
A
:
X
→
I
}
(7.8)
and
u
A
(
x
) is the membership degree of
x
to
A
.For
u
A
(
x
)=0,
x
does
not belong to
A
,andfor
u
A
(
x
)=1,
x
does belong to
A
. All other cases
are considered fuzzy.
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