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Also since
d
∗
(
≤
d
∗
(
A
1
,
A
2
)
≤
A
1
,
A
2
)
≤
0
1
(2.13)
then
λ
d
∗
(
λ
d
0
≤
A
1
,
A
2
)
−
∗
(
A
1
,
A
2
)
≤
1
,λ
≥
1
(2.14)
i.e.,
λ
d
∗
(
λ
d
0
≤
1
−
A
1
,
A
2
)
+
∗
(
A
1
,
A
2
)
λ
d
∗
(
λ
d
=
1
−
A
1
,
A
2
)
−
∗
(
A
1
,
A
2
)
≤
1
,λ
≥
1
(2.15)
ϑ(
Thus
A
1
,
A
2
)
is an IFV.
ϑ(
(2) If
A
1
,
A
2
)
=
(
1
,
0
)
, then
λ
d
∗
(
λ
d
1
−
A
1
,
A
2
)
=
1
,
∗
(
A
1
,
A
2
)
=
0
,λ
≥
1
(2.16)
Also since
λ
d
∗
(
λ
d
∗
(
1
=
1
−
A
1
,
A
2
)
≤
ϑ(
A
1
,
A
2
)
≤
1
−
A
1
,
A
2
),
λ
≥
1
(2.17)
i.e.,
ϑ(
A
1
,
A
2
)
=
1, by Eq. (
2.1
), we get
A
1
=
A
2
;otherwise,if
A
1
=
A
2
, then by
Eqs. (
2.8
) and (
2.9
), we have
ϑ(
A
1
,
A
2
)
=
(
1
,
0
)
.
ϑ(
A
2
)
=
ϑ(
A
1
,
A
2
,
A
1
)
(3) Obviously, we have
. This completes the proof of the
theorem.
Definition 2.2
(Zhang et al. 2007) Let
Z
=
(
z
ij
)
n
×
n
be a matrix, if all of its elements
z
ij
(
i
,
j
=
1
,
2
,...,
n
)
are IFVs, then
Z
is called an intuitionistic fuzzy matrix.
z
(
1
)
ij
z
(
2
)
ij
Definition 2.3
(Zhang et al. 2007) Let
Z
1
=
(
)
n
×
n
and
Z
2
=
(
)
n
×
n
be two
intuitionistic fuzzy matrices. If
Z
=
Z
1
◦
Z
2
, then
Z
is called the composition matrix
of
Z
1
and
Z
2
, where
n
∨
z
(
1
)
ik
z
(
2
)
kj
z
ij
=
1
(
∧
)
=
(
ma
k
{
min
{
μ
z
(
1
)
ik
,μ
z
(
2
)
kj
}
,
mi
k
{
max
{
v
z
(
1
)
ik
,
v
z
(
2
)
kj
}}
),
k
=
i
,
j
=
1
,
2
,...
n
(2.18)
Theorem 2.3
(Zhang et al. 2007) The composition matrix
Z
of the intuitionistic
fuzzy matrix
Z
1
and
Z
2
is also an intuitionistic fuzzy matrix.
z
(
1
)
ij
z
(
2
)
ij
Proof
Let
Z
1
=
(
)
n
×
n
,
Z
2
=
(
)
n
×
n
and
Z
=
(
z
ij
)
n
×
n
. Then by Eq. (
2.18
), we
have
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