Digital Signal Processing Reference
In-Depth Information
Definition 7.4:
Membership function of a crisp set
The fuzzy set
A
is called non ambiguous, or crisp, if
u
A
(
x
)
∈{
0
,
1
}
.
Definition 7.5:
Complement of a fuzzy set
If
A
is from
L
(
X
), the
complement
of A is the fuzzy set
A
, defined
as
u
A
(
x
)=1
−
u
A
(
x
)
,
∀
x
∈
X
(7.9)
In the following, we define fuzzy operations which allow us to work
with fuzzy sets defined by membership functions.
For two fuzzy sets
A
and
B
on
X
, the following operations can be
defined.
Definition 7.6:
Equality
Fuzzy set
A
is equal to fuzzy set
B
if and only if
u
A
(
x
)=
u
B
(
x
)for
all
X
.Insymbols,
A
=
B
⇐⇒
u
A
(
x
)=
u
B
(
x
)
,
∀
x
∈
X
(7.10)
The next two definitions are for the inclusion and the product of two
fuzzy sets.
Definition 7.7:
Inclusion
Fuzzy set
A
is contained in fuzzy set
B
if and only if
u
A
(
x
)
≤
u
B
(
x
)
for all
X
.Insymbols,
A
B
⇐⇒
u
A
(
x
)
≤
u
B
(
x
)
,
∀
x
∈
X
(7.11)
Definition 7.8:
Product
The product
AB
of fuzzy set
A
with fuzzy set
B
has a membership
function that is the product of the two separate membership functions.
In symbols,
u
(AB)
(
x
)=
u
A
(
x
)
·
u
B
(
x
)
,
∀
x
∈
X
(7.12)
Search WWH ::
Custom Search