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c
4
(
A
,
B
)
j
=
1
w
j
μ
A
(
x
j
)
x
j
)
·
μ
B
(
x
j
)
+
v
A
(
x
j
)
·
v
B
(
x
j
)
+
π
A
(
x
j
)
·
π
B
(
=
j
=
1
w
j
·
j
=
1
w
j
A
(
x
j
)
+
v
A
(
x
j
)
+
π
A
(
x
j
)
B
(
x
j
)
+
v
B
(
x
j
)
+
π
B
(
x
j
)
μ
μ
(2.164)
T
where
w
=
(
w
1
,
w
2
,...,
w
n
)
is the weight vector of
x
j
(
j
=
1
,
2
,...,
n
)
with
n
and
j
=
1
w
j
w
j
1. Similar to Eq. (
2.159
), Eq. (
2.164
)also
satisfies all the conditions of Definition 2.31.
If the universe of discourse,
X
, is continuous and theweight of the element
x
≥
0
,
j
=
1
,
2
,...,
=
∈
X
=
0 and
a
[
a
,
b
]
is
w
(
x
)
, where
w
(
x
)
≥
w
(
x
)
dx
=
1, then Eq. (
2.164
) is transformed
into the following form:
c
5
(
A
,
B
)
a
w
(
x
) (μ
A
(
x
)μ
B
(
x
)
+
v
A
(
x
)
v
B
(
x
)
+
π
A
(
x
)π
B
(
x
))
dx
=
a
dx
·
a
dx
(2.165)
2
A
(
x
)
+
v
A
(
x
)
+
π
2
2
B
(
x
)
+
v
B
(
x
)
+
π
2
w
(
x
)
μ
A
(
x
)
w
(
x
)
μ
B
(
x
)
1
If all the elements have the same importance, i.e.,
w
(
x
)
=
∈[
0
,
1
]
(in this
b
−
a
case,
(
b
−
a
)
≥
1
)
, for any
x
∈[
a
,
b
]
, then Eq. (
2.165
) is replaced by
a
(μ
A
(
x
)μ
B
(
x
)
+
v
A
(
x
)
v
B
(
x
)
+
π
A
(
x
)π
B
(
x
))
dx
c
6
(
A
,
B
)
=
a
μ
)
dx
·
a
μ
)
dx
(2.166)
2
v
A
(
2
2
v
B
(
2
A
(
x
)
+
x
)
+
π
A
(
x
B
(
x
)
+
x
)
+
π
B
(
x
2.7.2 Intuitionistic Fuzzy Clustering Algorithm
Let
C
is the association
coefficient of
A
i
and
A
j
, which is derived by one of the intuitionistic fuzzy association
measures (
2.157
) and (
2.162
)-(
2.164
). Then by Definition 2.12, we can directly
derive the following result:
Theorem 2.18
(Zhao et al. 2012b) Let
C
λ
=
λ
c
ij
m
×
m
be a
=
(
c
ij
)
m
×
m
be an association matrix, where
c
ij
=
c
(
A
i
,
A
j
)
λ
-cutting matrix of the
association matrix
C
c
ij
)
m
×
m
. Then
C
is an equivalent association matrix if and
only if
C
λ
is an equivalent Boole matrix, for all
=
(
λ
∈[
0
,
1
]
, that is,
⊆
C
if and only if
I
λ
⊆
⊆
(1)
C
is reflexive, i.e.,
I
C
λ
, i.e.,
I
C
λ
.
C
if and only if
C
T
(2)
C
is symmetric, i.e.,
C
T
T
=
λ
=
=
C
λ
, i.e.,
(
C
λ
)
C
λ
.
(3)
C
is transitive, i.e.,
C
2
⊆
C
if and only if
C
λ
◦
C
λ
⊆
C
.
λ
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