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In-Depth Information
•
Condition
(
VPUN1
)
with
ʲ
is equivalent to:
˅
A
(
)
=∅⃔
˅
C
(
u
u
)
=∅
for all u
∈
U
.
(VPUNG1)
The next lemma is the counterpart of Lemma
2
of RSM. However, only the
sufficient condition of the lemma holds in VPRSM.
∈
ʲ
∈[
,
.
)
Lemma 4
Let u
U be an object,
0
0
5
be an admissible error rate, and A
be a subset of C .
˅
A
(
)
=
˅
C
(
•
The following assertion is a sufficient condition of
u
u
)
:
,(˅
C
(
u
)
=
˅
C
(
u
∈
u
,
∀
U
u
)
⃒∃
a
∈
A
,(
u
)
∈
R
}
).
{
a
ʻ
A
(
)
=
ʻ
C
(
•
The following assertion is a sufficient condition of
u
u
)
:
,(ʻ
C
(
u
)
=
ʻ
C
(
u
∈
u
,
∀
U
u
)
⃒∃
a
∈
A
,(
u
)
∈
R
}
).
{
a
This lemma holds due to property (
7.9
). Then, we have the following corollary.
Corollary 1
We have the following equivalences:
,˅
A
(
)
=
˅
C
(
∀
u
∈
U
u
u
)
,(˅
C
(
u
)
=
˅
C
(
u
∈
u
,
⃔∀
,
)
⃒∃
∈
,(
)
∈
R
{
a
}
),
u
U
u
a
A
u
,ʻ
A
(
)
=
ʻ
C
(
∀
u
∈
U
u
u
)
,(ʻ
C
(
u
)
=
ʻ
C
(
u
∈
u
,
⃔∀
u
,
U
u
)
⃒∃
a
∈
A
,(
u
)
∈
R
{
a
}
).
It says that all L-reducts and all U-reducts can be enumerated by discernibility
functions. The similar result is shown in [
33
]. However, we do not have the same
result for
and conditions (
VPPG1
) and (
VPUNG1
).
We introduce a discernibility matrix
M
(˅
\
ʻ)
=
(
m
i
,
j
)
ij
=
1
,
2
,...,
n
, where
ij
-entry
m
ij
is
defined by:
m
ij
={
c
∈
C
|
a
(
u
i
)
=
a
(
u
j
)
}
.
It is the same as that of RSM. Then, we define discernibility functions corresponding
to L-reducts and U-reducts, which are denoted by
F
U
ʲ
and
F
L
ʲ
, respectively.
Definition 12
Let
ʲ
∈[
0
,
0
.
5
)
be an admissible error rate. Discernibility functions
F
U
ʲ
and
F
L
ʲ
are defined as follows:
F
U
ʲ
(
˜
c
1
,
˜
c
2
,...,
˜
c
m
)
=
m
ij
˜
c
,
c
∈
|
˅
C
(
u
i
)
=
˅
C
(
i
,
j
u
j
)
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