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n
∨
(3) Transitivity:
Z
2
⊆
Z
, i.e.,
1
(
z
ik
∧
z
kj
)
≤
z
ij
,
i
,
j
=
1
,
2
,...,
n
.
k
=
Then
Z
is called an intuitionistic fuzzy equivalence matrix.
In order to save computation, motivated by the idea of Wang (1983), we have the
following conclusion:
Theorem 2.6
(Zhang et al. 2007) Let
Z
be an intuitionistic fuzzy similarity matrix.
Then after the finite times of compositions:
Z
2
k
Z
2
Z
4
Z
→
→
→ ··· →
→ ···
There must exist a positive integer
k
such that
Z
2
k
Z
2
(
k
+
1
)
, and
Z
2
k
=
is an intuition-
istic fuzzy equivalence matrix.
Definition 2.6
(Zhang et al. 2007) Let
Z
=
(
z
ij
)
n
×
n
be an intuitionistic fuzzy simi-
larity matrix, where
z
ij
=
(μ
z
ij
,
v
z
ij
)
,
i
,
j
=
1
,
2
,...,
n
. Then
Z
λ
=
(
λ
z
ij
)
n
×
n
is called
the
λ
-cutting matrix of
Z
, where
⎧
⎨
0
,
if
λ>
1
−
v
z
ij
,
1
2
,
z
ij
=
(2.33)
if
μ
z
ij
<λ
≤
1
−
v
z
ij
,
λ
⎩
1
,
if
μ
z
ij
≥
λ.
Definition 2.7
(Wang 1983) If the matrix
Z
=
(
˙
z
ij
)
n
×
n
satisfies the following con-
ditions:
(1) Reflexivity:
z
ii
=
˙
1,
i
=
1
,
2
,...,
n
, and for any
z
ij
∈[
˙
0
,
1
]
,
i
,
j
=
1
,
2
,...,
n
.
(2) Symmetry:
z
ji
.
(3) Transitivity: ma
k
{
˙
z
ij
=˙
min
{˙
z
ik
,
˙
z
kj
}} ≤ ˙
z
ij
, for all
i
,
j
=
1
,
2
,...,
n
.
Then
Z
is called a fuzzy equivalence matrix.
Theorem 2.7
(Zhang et al. 2007)
Z
=
(
z
ij
)
n
×
n
is an intuitionistic fuzzy equivalence
matrix if and only if its
λ
-cutting matrix
Z
λ
=
(
λ
z
ij
)
n
×
n
is a fuzzy equivalence matrix,
where
z
ij
=
(μ
z
ij
,
v
z
ij
)
,
i
,
j
=
1
,
2
,...,
n
.
Proof
(Necessity)
(1)
(Reflexivity) Since
z
ii
=
(
1
,
0
)
,
λ
∈[
0
,
1
]
, then
λ
≤
μ
z
ii
=
1,
λ
z
ii
=
1.
(Symmetry) Since
z
ij
=
μ
z
ij
=
μ
z
ji
,
v
z
ij
=
v
z
ji
, thus
λ
z
ij
=
λ
z
ji
.
(2)
z
ji
, i.e.,
=
(
z
ij
)
n
×
n
is an intuitionistic fuzzy equivalence matrix,
(3)
(Transitivity) Since
Z
we have
ma
k
{
min
{
μ
z
ik
,μ
z
kj
}} ≤
μ
z
ij
(2.34)
mi
k
{
max
{
v
z
ik
,
v
z
kj
}} ≤
v
z
ij
(2.35)
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