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This definition is a direct generalization of what happens in the classical case. If
=
A
0
∪
A
1
∪···∪
A
n
, with
A
i
∩
= ∅
=
X
A
j
for
i
j
, is a classical partition of
X
,
it is
μ
A
0
(
x
)
+
μ
A
1
(
x
)
+···+
μ
A
n
(
x
)
=
1, since for each
x
∈
X
there is just one
A
j
such that
x
∈
A
j
,but
x
∈
A
i
,for
i
=
j
, that is,
μ
A
j
(
x
)
=
1
,
and
μ
A
i
(
x
)
=
0 f
i
=
j
,
that implies
j
=
0
μ
A
j
(
x
)
=
1. Let us show three examples.
=[
,
]
Example 1.5.1
In
X
0
4
,take
1
−
x
,
if
0
x
1
,
μ
0
(
x
)
=
0
if
1
x
4
,
⊧
⊨
0
,
if
0
x
j
−
1
,
x
+
1
−
j
if
j
−
1
x
j
,
μ
j
(
x
)
=
for
1
j
3
⊩
j
+
1
−
x
if
j
x
j
+
1
,
,
+
,
1
if
j
1
x
4
0
,
if
0
x
3
,
μ
4
(
)
=
x
x
−
3if
3
x
4
,
Graphically,
Obviously,
1,
j
=
0
μ
j
(
•
If 0
x
x
)
=
μ
0
(
x
)
+
μ
1
(
x
)
=
1
−
x
+
x
=
1
1,
j
=
0
μ
j
(
•
If 1
j
3
,
j
x
j
+
x
)
=
μ
j
(
x
)
+
μ
j
+
1
(
x
)
=
(
j
+
1
−
x
)
+
(
x
+
1
−
j
−
1
)
=
1
.
4,
j
=
0
μ
j
(
•
If 3
x
x
)
=
μ
3
(
x
)
+
μ
4
(
x
)
=
4
−
x
+
x
−
3
=
1
Hence
{
μ
0
, μ
1
,...,μ
n
}
is a fuzzy partition of
[
0
,
4
]
. Notice that each
μ
j
can be
=
μ
j
=
μ
A
j
.
labeled by the predicate around j
A
j
, that is
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