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In-Depth Information
Example 11
Remember the decision table
D =
(
U
,
C
∪{
d
}
,
{
V
a
}
)
in Table
7.7
.Let
=
{Ma, Ph}. The lower and upper approximations of the upward and downward
unions with respect to
A
are obtained as follows.
A
X
b
)
=
X
m
)
={
X
g
)
=∅
,
LA
A
(
U
,
LA
A
(
u
1
,
u
2
,
u
3
,
u
4
}
,
LA
A
(
X
b
)
={
X
m
)
={
X
g
)
=
LA
A
(
u
7
}
,
LA
A
(
u
4
,
u
5
,
u
6
,
u
7
}
,
LA
A
(
U
,
X
b
)
=
X
m
)
=
X
g
)
={
UA
A
(
U
,
UA
A
(
U
\{
u
7
}
,
UA
A
(
u
1
,
u
2
,
u
3
}
,
X
b
)
={
X
m
)
=
X
g
)
=
UA
A
(
u
5
,
u
6
,
u
7
}
,
UA
A
(
U
,
UA
A
(
U
.
Now, we remember properties of approximations [
16
,
18
,
31
]. By the boundary
conditions of
X
≥
and
X
≤
,
X
1
)
=
X
1
)
=
X
p
)
=
X
p
)
=
UA
A
(
LA
A
(
U
,
UA
A
(
LA
A
(
U
,
X
p
+
1
)
=
X
p
+
1
)
=∅
,
X
0
)
=
X
0
)
=∅
.
UA
A
(
LA
A
(
UA
A
(
LA
A
(
(7.21)
V
d
. Similarly to RSM, there exist inclusion relations between
each union of decision classes and its lower and upper approximations.
Let
A
ↆ
C
and
i
∈
X
i
X
i
X
i
X
i
X
i
X
i
LA
A
(
)
ↆ
ↆ
UA
A
(
),
LA
A
(
)
ↆ
ↆ
UA
A
(
).
(7.22)
Approximations are expressed by unions of dominating or dominated sets,
X
i
D
A
(
D
A
(
LA
A
(
)
=
u
)
=
u
),
D
A
(
X
i
X
i
u
)
ↆ
u
∈
LA
A
(
)
X
i
D
A
(
D
A
(
UA
A
(
)
=
u
)
=
u
),
D
A
(
X
i
=∅
X
i
u
)
∩
u
∈
UA
A
(
)
X
i
D
A
(
D
A
(
LA
A
(
)
=
u
)
=
u
),
D
A
(
X
i
X
i
u
)
ↆ
u
∈
LA
A
(
)
X
i
D
A
(
D
A
(
UA
A
(
)
=
u
)
=
u
).
D
A
(
u
)
∩
X
i
=∅
u
∈
UA
A
(
X
i
)
There exists duality of lower and upper approximations.
X
i
X
i
−
1
),
X
i
X
i
+
1
).
UA
A
(
)
=
U
\
LA
A
(
UA
A
(
)
=
U
\
LA
A
(
(7.23)
So, the upper approximations of the pair of complementary unions of decision classes
form a cover of
U
:
X
i
X
i
−
1
)
=
UA
A
(
)
∪
UA
A
(
U
.
(7.24)
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