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are monotone with respect to the inclusion relation between
condition attribute sets. Namely, for
B
l
A
(
u
)
and
u
A
(
u
)
,
ↆ
∈
A
C
and
u
U
,wehave
B
ↆ
A
⃒
l
B
(
u
)
≤
l
A
(
u
),
u
B
(
u
)
≥
u
A
(
u
).
(7.43)
V
d
, using the generalized decision function, the lower and upper approx-
imations of unions are represented as:
Let
i
∈
X
i
X
i
LA
A
(
)
={
u
∈
U
|
l
A
(
u
)
≥
i
}
,
UA
A
(
)
={
u
∈
U
|
u
A
(
u
)
≥
i
}
,
(7.44)
X
i
X
i
LA
A
(
)
={
u
∈
U
|
u
A
(
u
)
≤
i
}
,
UA
A
(
)
={
u
∈
U
|
l
A
(
u
)
≤
i
}
.
(7.45)
We can represent approximations of classes using the generalized decision,
LA
A
(
X
i
)
={
u
∈
U
|
l
A
(
u
)
=
u
A
(
u
)
=
i
}
,
(7.46)
UA
A
(
X
i
)
={
u
∈
U
|
l
A
(
u
)
≤
i
≤
u
A
(
u
)
}
,
(7.47)
BN
A
(
X
i
)
= {
u
∈
U
|
l
A
(
u
)
≤
i
≤
u
A
(
u
),
l
A
(
u
)<
u
A
(
u
)
}
.
(7.48)
Example 12
Remember the decision table
D =
(
U
,
C
∪{
d
}
,
{
V
a
}
)
in Table
7.7
.Let
A
=
{Ma, Ph}. The generalized decision function
ʴ
A
with respect to
A
is obtained
as follows:
ʴ
A
(
u
1
)
=
,
,ʴ
A
(
u
2
)
=
,
,ʴ
A
(
u
3
)
=
,
,
med
good
med
good
med
good
ʴ
A
(
u
4
)
=
med
,
med
,ʴ
A
(
u
5
)
=
bad
,
med
,ʴ
A
(
u
6
)
=
bad
,
med
,
ʴ
A
(
u
7
)
=
bad
,
bad
.
7.4.2 Structure-Based Reducts in Dominance-Based
Rough Set Models
Before defining structure-based reducts in DRSM, we introduce a notion of reducts
preserving the quality of sorting, proposed by Susmaga et al. [
49
]. For
A
ↆ
C
,the
quality of sorting
, which is the counterpart of the quality of classification in
the classical RSM, is defined by:
ʳ
A
(
d
)
−
i
∈
V
d
BN
A
(
−
i
∈
V
d
BN
A
(
X
i
X
i
)
=
|
U
)
|
=
|
U
)
|
ʳ
A
(
d
.
|
U
|
|
U
|
By (
7.34
) and (
7.40
), we can see that
ʳ
A
(
d
)
is related to the positive region of DRSM,
−
i
∈
V
d
BN
A
(
−
i
∈
V
d
BN
A
(
X
i
)
=
|
U
)
|
=
|
U
X
i
)
|
=
|
POS
A
(
d
)
|
ʳ
A
(
d
.
|
U
|
|
U
|
|
U
|
We call this type of reducts Q-reducts.
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