Information Technology Reference
In-Depth Information
=[
,
]
Example 1.5.2
In
X
0
10
,take
⊧
⊨
1
x
4
,
if
0
x
4
,
x
−
4
,
if
0
x
4
,
μ
0
(
x
)
=
,
μ
1
(
x
)
=
1
,
if
4
x
6
,
0
,
if
4
x
10
⊩
10
−
x
,
if
6
x
10
4
0
,
if
0
x
6
,
μ
2
(
x
)
=
x
−
6
,
if
6
x
10
.
4
Graphically
Since, obviously,
μ
0
(
x
)
+
μ
1
(
x
)
+
μ
2
(
x
)
=
1 for all
x
∈[
0
,
10
]
,
{
μ
0
, μ
1
, μ
2
}
is
a fuzzy partition of the interval
[
0
,
10
]
.
Example 1.5.3
By proceeding in the same way that in last example, it is easy to
prove that it is
{
μ
cold
, μ
hot
, μ
w
arm
}
,
a fuzzy partition of the interval
[−
10
,
50
]
.
Notice that, with the strong negation
N
0
=
1
−
id, if
{
μ
0
, μ
1
, μ
2
}
is a fuzzy partition
of
X
,itis
μ
1
+
μ
2
=
μ
0
μ
1
(
x
)
+
μ
2
(
x
)
=
1
−
μ
0
(
x
),
or
μ
0
+
μ
1
=
μ
2
μ
0
(
x
)
+
μ
1
(
x
)
=
1
−
μ
2
(
x
),
or
μ
0
+
μ
2
=
μ
1
μ
0
(
x
)
+
μ
2
(
x
)
=
1
−
μ
1
(
x
),
or
The complement (with
N
0
) of each element in a fuzzy partition, is just the addition
of the other elements in the partition.
1.6 A Note on Lattices
1. An algebraic structure
(
L
,
)
is a partially ordered set, or poset, if the relation
ↆ
L
×
L
is a partial order, that is, enjoys the properties:
•
Reflexive,
a
a
for all
a
in
L
.
•
Transitive,
(
a
b
)
&
(
b
c
)
⃒
(
a
c
)
.
•
(
)
(
)
⃔
=
Anti-symmetric,
a
b
&
b
a
a
b
.
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