Information Technology Reference
In-Depth Information
X ,itis
Notice that when
μ A ∈{
0
,
1
}
•[ μ ( 0 )
A
]=
X , since for all x
X it is 0
μ(
x
)
[ μ ( r )
A
If r
>
0,
]=
A , since for all x
A it is 0
<
r
1
= μ A (
x
),
then, the only r-cuts of a crisp subset A of X are X and A .
If r
[ μ ( s ) ]ↂ[ μ ( r ) ]
μ(
)
μ(
)
s , since s
x
implies r
x
,itresults
: r-cuts are
decreasing when their indices increase.
Let us show an example with X
={
1
,
2
,
3
,
4
,
5
}
and
μ =
0
.
8
/
1
+
0
.
6
/
2
+
0
.
7
/
3
+
1
5, where the only significative values for the r -cuts are 0.6, 0.7, 0.8, and 1:
•[ μ ( 0 . 6 ) ]={
/
4
+
1
/
1
,
2
,
3
,
4
,
5
}=
X
•[ μ ( 0 . 7 ) ]={
1
,
3
,
4
,
5
}
•[ μ ( 0 . 8 ) ]={
1
,
4
,
5
}
•[ μ ( 1 ) ]={
.
It is clear that 0
4
,
5
}
[ μ ( 1 ) ]ↂ[ μ ( 0 . 8 ) ]ↂ[ μ ( 0 . 7 ) ]ↂ[ μ ( 0 . 6 ) ]
.
6
0
.
7
0
.
8
1, and
.
X ,is
Theorem 2.1.1
(Theorem of resolution) Fo r a l l
μ ∈[
0
,
1
]
μ(
x
) =
max
{
r
), μ ( r ) (
[
0
,
1
];
min
r (
x
x
) }
, for all x
X.
Proof
), μ ( r ) (
, μ ( r ) (
0 r 1 min
max
r (
x
x
)) =
0 r 1 min
max
(
r
x
))
0 r 1 min r
1
r
,
if r
μ(
x
),
,
if r
μ(
x
),
=
max
,
=
max
0 r 1
= μ(
x
).
0
,
otherwise,
0
,
otherwise,
 
Search WWH ::




Custom Search