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The lower generalized decision function is a singleton or the empty set,
|
ʻ
A
(
u
)
|≤
1
.
(7.15)
Unlike RSM, we may have the case that the upper generalized decision of
u
is
a singleton, i.e.,
˅
A
(
u
)
={
i
}
but
u
does not belong to the lower approximation
LA
A
(
. When the lower generalized decision is a singleton, the upper generalized
decision is also a singleton, and they are the same,
X
i
)
|
ʻ
A
(
⃒
˅
A
(
)
=
ʻ
A
(
u
)
|=
1
u
u
).
(7.16)
Property (
7.12
) can be expressed as:
1
p
>ʲ
⃒
˅
A
(
u
)
=∅
.
(7.17)
˅
A
(
is less than
p
.
)
ʲ
Hence,
u
may be empty unless
(˅
\
ʻ)
A
(
We define a function
u
)
as:
(˅
\
ʻ)
A
(
)
=
˅
A
(
)
\
ʻ
A
(
u
u
u
).
By properties (
7.13
), (
7.15
), and (
7.16
), we have
(˅
\
ʻ)
A
(
)
=∅⃒
˅
A
(
ʻ
A
(
u
u
)
=∅
or
u
)
=∅
,
(7.18)
(˅
\
ʻ)
A
(
)
=∅⃒
(˅
\
ʻ)
A
(
)
=
˅
A
(
u
u
u
).
(7.19)
By that property, the following equivalence holds:
∈
(˅
\
ʻ)
A
(
BN
A
(
i
u
)
⃔
u
∈
X
i
).
(7.20)
Therefore, we call
(˅
\
ʻ)
a boundary generalized decision function.
Example 8
Remember the decision table
D =
(
U
,
C
∪{
d
}
,
{
V
a
}
)
in Table
7.3
.Let
ʲ
=
0
.
39. The lower and upper generalized decision function with respect to
C
and
ʲ
are,
ʻ
C
(
}
,ʻ
C
(
}
,ʻ
C
(
P
3
)
=∅
,ʻ
C
(
C
(
P
1
)
={
g
P
2
)
={
m
P
4
)
=∅
,
P
5
)
=∅
,
˅
C
(
}
,˅
C
(
}
,˅
C
(
}
,˅
C
(
}
,˅
C
(
P
1
)
={
P
2
)
={
P
3
)
={
P
4
)
={
P
5
)
=∅
,
g
m
g
m,g
ʻ
C
(
˅
C
(
where
P
i
)
and
P
i
)
indicate the lower and upper generalized decisions of an
object in the group
P
i
.
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