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(α
1
∧
α
2
)
∨
α
3
=
(
{
{
μ
α
1
,μ
α
2
}
,μ
α
3
}
,
{
{
v
α
1
,
v
α
2
}
,
v
α
3
}
)
(2)
max
min
min
max
=
(
{
{
μ
α
1
,μ
α
3
}
,
{
μ
α
2
,μ
α
3
}}
,
min
max
max
max
{
min
{
v
α
1
,
v
α
3
}
,
min
{
v
α
2
,
v
α
3
}}
)
=
(α
1
∨
α
3
)
∧
(α
2
∨
α
3
)
(3)
(α
1
∨
α
2
)
∨
α
3
=
(
max
{
max
{
μ
α
1
,μ
α
2
}
,μ
α
3
}
,
min
{
min
{
v
α
1
,
v
α
2
}
,
v
α
3
}
)
=
(
max
{
μ
α
1
,μ
α
2
,μ
α
3
}
,
min
{
v
α
1
,
v
α
2
,
v
α
3
}
)
=
(
max
{
μ
α
1
,
max
{
μ
α
2
,μ
α
3
}
,
min
{
v
,
min
{
v
,
v
}
)
α
α
α
1
2
3
=
α
1
∨
(α
2
∨
α
3
)
(4)
(α
1
∧
α
2
)
∧
α
3
=
(
min
{
min
{
μ
α
1
,μ
α
2
}
,μ
α
3
}
,
max
{
max
{
v
,
v
}
,
v
}
)
α
α
α
1
2
3
=
(
min
{
μ
α
1
,μ
α
2
,μ
α
3
}
,
max
{
v
,
v
,
v
}
)
α
α
α
1
2
3
=
(
min
{
μ
α
1
,
min
{
μ
α
2
,μ
α
3
}
,
max
{
v
,
max
{
v
,
v
}
)
α
α
α
1
2
3
=
α
1
∧
(α
2
∧
α
3
)
which completes the proof.
Let
X
={
x
1
,
x
2
,...,
x
n
}
be a finite universe of discourse,
A
1
={
x
i
,μ
A
1
(
x
i
),
v
A
1
(
be two IFSs. Atanassov (1983,
1986) suggested the inclusion relations between the IFSs as follows:
x
i
)
|
x
i
∈
X
}
and
A
2
={
x
i
,μ
A
2
(
x
i
),
v
A
2
(
x
i
)
|
x
i
∈
X
}
(1)
A
1
⊆
A
2
if and only if
μ
A
1
(
x
i
)
≤
μ
A
2
(
x
i
)
and
v
A
1
(
x
i
)
≥
v
A
2
(
x
i
)
, for any
x
i
∈
X
;
(2)
A
1
=
A
2
if and only if
A
1
⊆
A
2
and
A
1
⊇
A
2
, i.e.,
μ
A
1
(
x
i
)
=
μ
A
2
(
x
i
)
and
v
A
1
(
x
i
)
=
v
A
2
(
x
i
)
, for any
x
i
∈
X
.
In fuzzy mathematics, the similarity matrix with reflexivity and symmetry is a
common matrix. Zhang et al. (2007) introduced the similarity matrix to the IFS
theory, and defined the concept of intuitionistic fuzzy similarity degree:
ϑ
:
(
2
Definition 2.1
(Zhang et al. 2007) Let
IFS
(
X
))
→
IFS
(
X
)
, where IFS
(
X
)
.If
ϑ(
indicates the set of all IFSs, and let
A
i
∈
IFS
(
X
) (
i
=
1
,
2
,
3
)
A
1
,
A
2
)
satisfies
the following properties:
ϑ(
(1)
A
1
,
A
2
)
is an IFV.
ϑ(
(2)
A
1
,
A
2
)
=
(
1
,
0
)
if and only if
A
1
=
A
2
.
ϑ(
A
2
)
=
ϑ(
(3)
A
1
,
A
2
,
A
1
)
.
Then
ϑ(
A
1
,
A
2
)
is called an intuitionistic fuzzy similarity degree of
A
1
and
A
2
.
Liu (2005) gave a formula for calculating the similarity degree between
A
1
and
A
2
:
n
w
i
β
1
|
μ
A
1
(
ϑ(
x
i
)
|
λ
+
β
2
|
x
i
)
|
λ
A
1
,
A
2
)
=
−
x
i
)
−
μ
A
2
(
v
A
1
(
x
i
)
−
v
A
2
(
1
i
=
1
1
λ
x
i
)
|
λ
+
β
3
|
π
A
1
(
x
i
)
−
π
A
2
(
(2.1)
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