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)
A
(
1
={
x
,
˜
v
A
(
x
), μ
A
(
x
)
|
x
∈
X
}
.
)
A
1
∩
A
2
={
(
2
x
,
[
min
{
inf
μ
A
1
(
x
),
inf
μ
A
2
(
x
)
}
,
min
{
sup
μ
A
1
(
x
),
sup
μ
A
2
(
x
)
}]
,
[
max
{
inf
ν
A
1
(
x
),
inf
v
˜
A
2
(
x
)
}
,
max
{
sup
v
˜
A
1
(
x
),
sup
v
˜
A
2
(
x
)
}] |
x
∈
X
}
.
)
A
1
∪
A
2
={
(
3
x
,
[
max
{
inf
μ
A
1
(
x
),
inf
μ
A
2
(
x
)
}
,
max
{
sup
μ
A
1
(
x
),
sup
μ
A
2
(
x
)
}]
,
[
min
{
inf
˜
v
A
1
(
x
),
inf
v
˜
A
2
(
x
)
}
,
min
{
sup
v
˜
A
1
(
x
)),
sup
˜
v
A
2
(
x
)
]}|
x
∈
X
}
.
)
A
1
+
A
2
={
(
4
x
,
[
inf
μ
A
1
(
x
)
+
inf
μ
A
2
(
x
)
−
inf
μ
A
1
(
x
)
·
inf
μ
A
2
(
x
),
sup
μ
A
1
(
x
)
+
sup
μ
A
2
(
x
)
−
sup
μ
A
1
(
x
)
·
sup
μ
A
2
(
x
)
]
,
[
inf
˜
v
A
1
(
x
)
·
inf
˜
v
A
2
(
x
),
sup
v
˜
A
1
(
x
)
·
sup
v
˜
A
2
(
x
)
]|
x
∈
X
}
.
)
A
1
·
A
2
={
(
5
x
,
[
inf
μ
A
1
(
x
)
·
inf
μ
A
2
(
x
),
sup
μ
A
1
(
x
)
·
sup
μ
A
2
(
x
)
]
,
[
inf
˜
v
A
1
(
x
)
+
inf
˜
v
A
2
(
x
)
−
inf
˜
v
A
1
(
x
).
inf
˜
v
A
2
(
x
),
˜
v
A
1
(
)
+
˜
v
A
2
(
)
−
˜
v
A
1
(
)
·
v
A
2
(
˜
)
]|
∈
}
.
sup
x
sup
x
sup
x
sup
x
x
X
Taking into account the needs of the application, Xu and Chen (2007a) further intro-
duced another two operational laws:
λ
A
))
λ
,
))
λ
]
,
(6)
={
x
,
[
1
−
(
1
−
inf
μ
A
(
x
1
−
(
1
−
sup
μ
A
(
x
))
λ
,(
))
λ
]|
[
(
inf
˜
v
A
(
x
sup
v
˜
A
(
x
x
∈
X
}
,λ >
0
.
A
λ
={
))
λ
,(
))
λ
]
,
(7)
x
,
[
(
inf
μ
A
(
x
sup
μ
A
(
x
))
λ
,
))
λ
]|
[
1
−
(
1
−
inf
v
A
(
˜
x
1
−
(
1
−
sup
˜
v
A
(
x
x
∈
X
}
,λ >
0
.
A
1
={
={
x
1
,
x
2
,...,
x
n
}
x
i
, μ
A
1
(
x
i
)
Let
X
be a discrete universe of discourse,
,
and
A
2
={
v
A
1
(
˜
x
i
)
|
x
i
∈
X
}
x
i
, μ
A
2
(
x
i
),
˜
v
A
2
(
x
i
)
|
x
i
∈
X
}
two IVIFSs, where
x
i
)
=[
μ
A
1
(
x
i
), μ
A
1
(
x
i
)
=[
μ
A
2
(
x
i
), μ
A
2
(
μ
A
1
(
x
i
)
]
, μ
A
2
(
x
i
)
]
(2.96)
v
A
1
(
v
A
1
(
v
A
2
(
v
A
2
(
˜
v
A
1
(
x
i
)
=[
x
i
),
x
i
)
]
,
˜
v
A
2
(
x
i
)
=[
x
i
),
x
i
)
]
(2.97)
Atanassov and Gargov (1989) defined the inclusion relation between two IVIFSs:
A
1
⊆
A
2
if and only if
μ
A
1
(
x
i
)
≤
μ
A
2
(
x
i
), μ
A
1
(
x
i
)
≤
μ
A
2
(
v
A
1
(
v
A
2
(
(1)
x
i
),
x
i
)
≥
x
i
)
and
v
A
1
(
v
A
2
(
x
i
)
≥
x
i
),
x
i
∈
X
.
A
1
=
A
2
if and only if
A
1
⊆
A
2
A
1
⊇
A
2
.
(2)
Similar to Definition 2.9, we have
A
j
Definition 2.15
(Xu et al. 2008) Let
(
j
=
1
,
2
,...,
m
)
be
m
IVIFSs. Then
C
(
A
i
,
A
j
)
is the asso-
ciation coefficient of
A
i
and
A
j
(which can be derived by one of the interval-valued
intuitionistic fuzzy association measures introduced by Xu and Chen (2008)), and
has the following properties:
=
(
˙
c
ij
)
m
×
m
is called an association matrix, where
˙
c
ij
=
c
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