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>
what follows is that, from the equivalence given by the existence of
r
0 such that
(
,
)>
(
,
)>
E
x
y
r
,
E
y
z
r
,itis
)
=
˕
−
1
W
˕
(
r
,
r
(
max
(
0
,
2
˕(
r
)
−
1
))
≤
W
˕
(
E
(
x
,
y
),
E
(
y
,
z
))
≤
E
(
x
,
z
).
Hence, to have 0
<
E
(
x
,
z
)
, it is necessary that 0
<
W
˕
(
r
,
r
)
, that is,
> ˕
−
1
0
<
max
(
0
,
2
˕(
r
)
−
1
)
,or
r
(
0
.
5
).
x
2
it
For example, if i
t is
T
=
W
, it should be
r
>
0
.
5, and if
T
=
W
with
˕(
x
)
=
˕
√
0
>
.
=
.
should be
r
5
0
7071. In the case of the fuzzy relation
i
−
1
min
(μ(
x
i
), ˃(
x
i
))
E
(μ, ˃)
=
(
i
−
1
μ(
x
i
),
i
−
1
˃(
x
i
))
,
max
x
2
. Thus, if
with
X
={
x
1
,...,
x
n
}
,itresults
W
-transitive for
˕(
x
)
=
˕
0
.
71
<
E
(μ, ˃)
, and 0
.
71
<
E
(˃, ʱ)
max
71
2
it follows 0
<
E
(μ, ʱ)
, since
W
˕
(
0
.
71
,
0
.
71
)
=
(
0
,
2
×
0
.
−
1
)
=
0
.
09.
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