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x
i
)
|
λ
+
β
2
|
≥
mi
i
{
β
1
|
μ
A
1
(
x
i
)
−
μ
A
2
(
v
A
1
(
x
i
)
x
i
)
|
λ
+
β
3
|
π
A
1
(
x
i
)
|
λ
}
,λ
≥
−
v
A
2
(
x
i
)
−
π
A
2
(
1
(2.6)
There must exist a positive integer
s
such that
x
i
)
|
λ
+
β
2
|
x
i
)
|
λ
+
β
3
|
π
A
1
(
x
i
)
|
λ
}
mi
i
{
β
1
|
μ
A
1
(
x
i
)
−
μ
A
2
(
v
A
1
(
x
i
)
−
v
A
2
(
x
i
)
−
π
A
2
(
x
s
)
|
λ
+
β
2
|
x
s
)
|
λ
+
β
3
|
π
A
1
(
=
β
1
|
μ
A
1
(
x
s
)
−
μ
A
2
(
v
A
1
(
x
s
)
−
v
A
2
(
x
s
)
x
s
)
|
λ
}
,λ
≥
−
π
A
2
(
1
(2.7)
As a result, when
w
s
=
1 and
w
i
=
0,
i
=
s
, the equality holds. Let
x
i
)
|
λ
+
β
2
|
x
i
)
|
λ
d
∗
(
A
1
,
A
2
)
=
mi
i
{
β
1
|
μ
A
1
(
x
i
)
−
μ
A
2
(
v
A
1
(
x
i
)
−
v
A
2
(
x
i
)
|
λ
}
,λ
≥
+
β
3
|
π
A
1
(
x
i
)
−
π
A
2
(
1
(2.8)
d
∗
(
x
i
)
|
λ
+
β
2
|
x
i
)
|
λ
A
1
,
A
2
)
=
ma
i
{
β
1
|
μ
A
1
(
x
i
)
−
μ
A
2
(
v
A
1
(
x
i
)
−
v
A
2
(
x
i
)
|
λ
}
,λ
≥
+
β
3
|
π
A
1
(
x
i
)
−
π
A
2
(
1
(2.9)
Thus
λ
d
∗
(
λ
d
∗
(
A
2
)
≤
ϑ
(
1
−
A
1
,
A
1
,
A
2
)
≤
1
−
A
1
,
A
2
), λ
≥
1
(2.10)
Based on Eqs. (
2.8
) and (
2.9
), Zhang et al. (2007) gave a formula for calculating
the similarity degree between two IFSs:
Theorem 2.2
(Zhang et al. 2007) Let
A
1
and
A
2
be two IFSs. Then
1
λ
d
∗
(
λ
d
∗
(
ϑ(
A
1
,
A
2
)
=
−
A
1
,
A
2
),
A
1
,
A
2
)
,λ
≥
1
(2.11)
is called the similarity degree between
A
1
and
A
2
.
ϑ(
Proof
(1) We first prove that
A
1
,
A
2
)
is an IFV. Since
x
i
)
|
λ
+
β
2
|
x
i
)
|
λ
+
β
3
|
π
A
1
(
x
i
)
|
λ
≤
β
1
|
μ
A
1
(
x
i
)
−
μ
A
2
(
v
A
1
(
x
i
)
−
v
A
2
(
x
i
)
−
π
A
2
(
0
x
i
)
|
λ
,
|
x
i
)
|
λ
,
≤
(β
1
+
β
2
+
β
3
)
max
{|
μ
A
1
(
x
i
)
−
μ
A
2
(
v
A
1
(
x
i
)
−
v
A
2
(
x
i
)
|
λ
}
|
π
A
1
(
x
i
)
−
π
A
2
(
x
i
)
|
λ
,
|
x
i
)
|
λ
,
|
π
A
1
(
x
i
)
|
λ
}≤
=
{|
μ
A
1
(
x
i
)
−
μ
A
2
(
v
A
1
(
x
i
)
−
v
A
2
(
x
i
)
−
π
A
2
(
,
λ
≥
max
1
1
then
λ
d
∗
(
λ
d
0
≤
1
−
A
1
,
A
2
)
≤
1
,
0
≤
∗
(
A
1
,
A
2
)
≤
1
,λ
≥
1
(2.12)
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