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(
1
,
0
),
A
=
B
,
R
(
A
,
B
)
=
(2.125)
c
(
A
·
B
)
∩
(
A
◦
B
)
,
A
=
B
,
then
R
(
A
,
B
)
is called the closeness degree of
A
and
B
.
By Eq. (
2.125
), we have
Theorem 2.15
(Xu et al. 2011) The closeness degree
R
(
A
,
B
)
of
A
and
B
is an
intuitionistic fuzzy similarity relation.
Proof
(1) We first prove that
R
(
A
,
B
)
is an IFV, since
A
and
B
are two IFSs on
X
,
we have
(a)
If
A
=
B
, then
R
(
A
,
B
)
=
(
1
,
0
)
;
(b)
If
A
=
B
, then
0
≤
μ
A
(
x
),
v
A
(
x
)
≤
1
,
0
≤
μ
A
(
x
)
+
v
A
(
x
)
≤
1
(2.126)
0
≤
μ
B
(
x
),
v
B
(
x
)
≤
1
,
0
≤
μ
B
(
x
)
+
v
B
(
x
)
≤
1
(2.127)
c
(
A
◦
B
)
={
X
(
v
A
(
x
)
∧
v
B
(
x
)),
X
(μ
A
(
x
)
∨
μ
B
(
x
))
}
(2.128)
R
(
A
,
B
)
=
(
min
{
X
(μ
A
(
x
)
∨
μ
B
(
x
)),
X
(
v
A
(
x
)
∧
v
B
(
x
))
}
,
min
{
X
(
v
A
(
x
)
∨
v
B
(
x
)),
X
(μ
A
(
x
)
∨
μ
B
(
x
))
}
)
(2.129)
(
,
)
Thus,
R
A
B
is an IFV.
(2) Since
R
(
A
,
A
)
=
(
1
,
0
)
, then
R
is reflexive.
c
c
(3) Since
R
(
A
,
B
)
=
(
A
·
B
)
∧
(
A
◦
B
)
=
(
B
·
A
)
∧
(
B
◦
A
)
=
R
(
B
,
A
)
, then
R
is
symmetrical. Thus,
R
(
A
,
B
)
is an intuitionistic fuzzy similarity relation.
Definition 2.24
(Xu et al. 2011) Let
R
=
(
r
ij
)
n
×
n
be an intuitionistic fuzzy similarity
matrix, where
r
ij
=
(μ
ij
,
v
ij
),
i
,
j
=
1
,
2
,...,
n
.
Then
R
=
(
(λ,δ)
r
ij
)
n
×
n
=
(λ,δ)
(
λ
μ
ij
,
δ
v
ij
)
n
×
n
is called a
(λ, δ)
-cutting matrix of
R
, where
(λ, δ)
is the confidence
level, 0
≤
λ, δ
≤
1
,
0
≤
λ
+
δ
≤
1, and
(
1
,
0
),
if
μ
ij
≥
λ,
v
ij
≤
δ,
r
ij
=
(
λ
μ
ij
,
δ
v
ij
)
=
(2.130)
(λ,δ)
(
0
,
1
),
if
μ
ij
<λ,
v
ij
>δ.
Theorem 2.16
(Xu et al. 2011)
R
=
(
r
ij
)
n
×
n
is an intuitionistic fuzzy similarity
matrix if and only if its
(λ, δ)
-cutting matrix
R
=
(
(λ,δ)
r
ij
)
n
×
n
is an intuitionistic
(λ,δ)
fuzzy similarity matrix.
Proof
r
ij
)
n
×
n
is an intuitionistic fuzzy similarity matrix, then
(1) (Reflexivity) Since
r
ii
(Necessity) If
R
=
(
=
(
1
,
0
),
0
≤
λ, δ
≤
1
,
0
≤
λ
+
δ
≤
1
,
then
r
ij
=
(
,
)
1
0
.
(λ,δ)
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