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Approximations of decision classes have similar properties as those of unions of
decision classes:
LA
A
(
X
i
)
ↆ
X
i
ↆ
UA
A
(
X
i
),
(7.35)
UA
A
(
X
i
)
=
BN
A
(
X
i
)
∪
X
i
,
(7.36)
LA
A
(
X
i
)
=
X
i
\
BN
A
(
X
i
).
(7.37)
The next properties are analogies to (
7.6
) and (
7.5
) of the classical RSM.
BN
A
(
X
i
)
=
UA
A
(
X
i
)
∩
UA
A
(
X
j
),
(7.38)
j
=
i
LA
A
(
X
i
)
=
U
\
UA
A
(
X
j
).
(7.39)
j
=
i
We define the positive region for the decision table in DRSM:
POS
A
(
d
)
=
LA
A
(
X
i
).
i
∈
V
d
The complement of the positive region is exactly the union of all boundaries,
U
\
POS
A
(
d
)
=
BN
A
(
X
i
).
(7.40)
i
∈
V
d
Moreover, the approximations are also monotone with respect to the inclusion
relation between condition attribute sets. Let
A
,
B
ↆ
C
and
i
∈
V
d
.
B
ↆ
A
⃒
LA
B
(
X
i
)
ↆ
LA
A
(
X
i
),
UA
B
(
X
i
)
ↇ
UA
A
(
X
i
).
(7.41)
The generalized decision function proposed by Dembczynski et al. [
10
] also plays
an important role for Boolean reasoning in DRSM. It provides an object-wise view
of DRSM. Let
A
ↆ
C
and
u
∈
U
, generalized decision of
u
with respect to
A
is
defined by
ʴ
A
(
u
)
=
l
A
(
u
),
u
A
(
u
)
, where
D
A
(
l
A
(
u
)
=
min
{
i
∈
V
d
|
u
)
∩
X
i
=∅}
,
D
A
(
u
A
(
u
)
=
max
{
i
∈
V
d
|
u
)
∩
X
i
=∅}
.
ʴ
A
(
u
)
shows the interval of decision classes to which
x
may belong.
l
A
(
u
)
and
u
A
(
u
)
are the lower and upper bounds of the interval. Obviously, we have
l
A
(
u
)
≤
u
A
(
u
).
(7.42)
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