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(
J
T
(
,
),
J
T
(
,
))
≤
J
T
(
,
)
,
,
[
,
]
T
a
b
b
c
a
c
for all
a
b
c
in
0
1
To avoid some difficult technicalities, we will exemplify this general result in the
particular case
J
W
(
a
,
b
)
=
min
(
1
,
1
−
a
+
b
)
:
W
(
J
W
(
a
,
b
),
J
W
(
b
,
c
))
=
W
(
min
(
1
,
1
−
a
+
b
),
min
(
1
,
1
−
b
+
c
))
=
(
,
(
,
b
−
a
)
+
(
,
−
b
+
c
))
max
0
min
0
min
1
1
min
(
1
,
1
−
b
+
c
)
≤
(
1
,
1
−
a
+
c
),
if
a
>
b
=
max
(
0
,
b
−
a
+
min
(
1
,
1
−
b
+
c
))
≤
min
(
1
,
1
−
a
+
c
),
if
a
≤
b
≤
J
W
(
a
,
c
).
Then, with the T-transitivity of
J
T
,itis
T
(
E
T
(
a
,
b
),
E
T
(
b
,
c
))
=
T
(
min
(
J
T
(
a
,
b
),
J
T
(
b
,
a
)),
min
(
J
T
(
b
,
c
),
J
T
(
c
,
b
)))
≤
T
(
J
T
(
a
,
b
),
J
(
b
,
c
))
=
E
T
(
a
,
c
),
hence, all relations
E
T
are T-transitive. Then,
- All relations
J
T
are T-Preorders, and
- All relations
E
T
are T-Indistinguishabilities
•
If
{
Ri
;
i
∈
I
}
is a collection of T-Preorders
,
Inf
i
∈
I
R
i
(
x
,
y
)
=
R
(
x
,
y
)
,
is also a
T-Preorder
.
Obviously,
R
(
x
,
x
)
=
Inf
i
∈
I
R
i
(
x
,
x
)
=
1, and
(
(
,
),
(
,
))
=
(
R
i
(
,
),
R
i
(
,
))
T
R
a
b
R
b
c
T
Inf
i
∈
I
a
b
Inf
i
∈
I
b
c
≤
Inf
i
T
(
R
i
(
a
,
b
),
R
i
(
b
,
c
))
∈
I
≤
R
i
(
,
)
=
(
,
),
Inf
i
a
c
R
a
c
∈
I
since T is a
continuous
t-norm.
•
If
{
E
i
;
i
∈
I
}
is a collection of T-indistinguishabilities
,
Inf
i
∈
I
E
i
(
x
,
y
)
=
E
(
x
,
y
)
is
also a T-indistinguishability
.
Obviously,
E
(
a
,
a
)
=
Inf
i
E
i
(
a
,
a
)
=
1
,
∈
I
and
E
(
a
,
b
)
=
Inf
i
E
i
(
a
,
b
)
=
Inf
i
E
i
(
b
,
a
)
=
E
(
b
,
a
).
∈
I
∈
I
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