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Table 7.3
An example of
decision table
c
1
c
2
c
3
c
4
d
:
(b, m, g)
P
1
Good
Good
Bad
Good
(0, 2, 9)
P
2
Good
Good
Good
Bad
(0, 19, 1)
P
3
Bad
Good
Bad
Good
(1, 1, 2)
P
4
Bad
Bad
Bad
Good
(0, 1, 1)
P
5
Good
Bad
Good
Good
(1, 1, 1)
respectively. In Table
7.3
, objects are classified into 5 groups
P
1
,
P
5
by the
condition attributes
C
. For example, group
P
1
is composed of objects having a con-
dition attribute tuple
P
2
,...,
(
c
1
,
c
2
,
c
3
,
c
4
)
=
(good, good, bad, good)
∈
V
C
. The number
of objects in each class in each group is shown in column
d
(b,m,g)inTable
7.3
.
For example, (0,2,9) of group
P
1
means that no object is in class
X
b
, 2 objects are
in class
X
m
and 9 objects are in class
X
g
.
The rough membership of an object
u
in each group
P
i
to each class
X
k
with
respect to
C
is the number of objects in
P
i
and
X
k
divided by the number of objects
in
P
i
. For example,
:
C
C
μ
X
b
(
P
1
)
=
0
/(
0
+
2
+
9
)
=
0,
μ
X
m
(
P
1
)
=
2
/(
0
+
2
+
9
)
=
C
0
.
1818
...
,
μ
X
g
(
P
1
)
=
9
/(
0
+
2
+
9
)
=
0
.
8181
...
. Given a condition attribute
subset
A
, the objects in
P
1
and
P
2
are indiscernible to each other. Hence,
the rough membership of an object
u
in
P
1
and
P
2
with respect to
A
becomes
μ
={
c
1
,
c
2
}
A
A
A
A
X
b
(
P
1
)
=
μ
X
b
(
P
2
)
=
/(
+
+
)
=
μ
X
m
(
P
1
)
=
μ
X
m
(
P
2
)
=
/(
+
0
0
21
10
0,
21
0
A
A
21
+
10
)
=
0
.
6774
...
,
μ
X
g
(
P
1
)
=
μ
X
g
(
P
2
)
=
10
/(
0
+
21
+
10
)
=
0
.
3225
...
.
39. The lower approximations and the upper approximations with
respect to
C
and
Let
ʲ
=
0
.
ʲ
are obtained as follows:
LA
C
(
UA
C
(
X
b
)
=∅
,
X
b
)
=∅
,
LA
C
(
UA
C
(
X
m
)
={
P
2
}
,
X
m
)
={
P
2
,
P
4
}
,
LA
C
(
UA
C
(
P
4
}
,
where we express approximations by means of groups, namely, all members of a
group
P
are members of an approximation
X
if
P
X
g
)
={
P
1
}
,
X
g
)
={
P
1
,
P
3
,
∈
X
.
InVPRSM, the properties corresponding to (
7.5
) and (
7.6
) are not always satisfied.
Consequently, L-reducts, U-reducts, and B-reducts become independent concepts in
VPRSM, and there are no strong-weak relations among them.
Additionally, property (
7.7
) only partially holds:
1
p
>ʲ
⃒
UA
A
(
=
X
i
).
U
(7.12)
i
∈
V
d
The union of upper approximations of all decision classes does not always equal
to
U
but when 1
/
p
>ʲ
. From this fact we define an unpredictable region of
d
with
and
A
, denoted by UNP
A
(
respect to
ʲ
d
)
, as follows:
UNP
A
(
NEG
A
(
d
)
=
X
i
),
i
∈
V
d
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