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F
UN
ʲ
(
˜
c
1
,
˜
c
2
,...,
˜
c
m
)
=
m
ij
˜
c
,
c
∈
UN
ʲ
(
i
,
j
)
∈
ʔ
where,
⊧
⊨
(˅
\
ʻ)
C
(
u
i
)
=
(˅
\
ʻ)
C
(
u
j
),
or
(˅
\
ʻ)
C
(
u
i
)
=
(˅
\
ʻ)
C
(
B
u
j
)
=∅
(
i
,
j
)
∈
ʔ
ʲ
⃔
⊩
(˅
\
ʻ)
C
\
m
ij
(
u
i
)
=
(˅
\
ʻ)
C
\
m
ij
(
and
u
j
)
=∅
,
⊧
⊨
ʻ
C
(
ʻ
C
(
u
i
)
=∅
and
u
j
)
=∅
,
or
ʻ
C
(
ʻ
C
(
P
(
i
,
j
)
∈
ʔ
ʲ
⃔
u
i
)
=∅
and
u
j
)
=∅
,
or
⊩
ʻ
C
(
u
i
)
=∅
,ʻ
C
(
ʻ
C
\
m
ij
(
u
i
)
=
ʻ
C
\
m
ij
(
u
j
)
=∅
,
and
u
j
)
=∅
,
⊧
⊨
˅
C
(
˅
C
(
u
i
)
=∅
and
u
j
)
=∅
,
or
˅
C
(
˅
C
(
UN
(
i
,
j
)
∈
ʔ
ʲ
⃔
u
i
)
=∅
and
u
j
)
=∅
,
or
⊩
˅
C
(
u
i
)
=∅
,˅
C
(
˅
C
\
m
ij
(
u
i
)
=
˅
C
\
m
ij
(
u
j
)
=∅
,
and
u
j
)
=∅
.
Proposition 3
Let A be a subset of C , and
ʲ
∈[
0
,
0
.
5
)
be an admissible error rate.
We have the following implications:
If F
B
c
A
•
ʲ
(
˜
)
=
0
then A does not satisfy
(
VPB1
)
or
(
VPB2
)
with
ʲ
,
If F
P
c
A
•
ʲ
(
˜
)
=
0
then A does not satisfy
(
VPP1
)
or
(
VPP2
)
with
ʲ
,
If F
UN
ʲ
c
A
•
(
˜
)
=
0
then A does not satisfy
(
VPUN1
)
or
(
VPUN2
)
with
ʲ
.
From Proposition
3
, we know that any prime implicant of each of
F
B
ʲ
,
F
P
ʲ
, and
F
UN
ʲ
can be a subset of some reduct of the corresponding type.
Combining Propositions
2
and
3
, we have the following theorem.
Theorem 6
Let A be a subset of C , and
ʲ
∈[
0
,
0
.
5
)
be an admissible error rate.
P
P
P
B
ʲ
P
ʲ
UN
ʲ
Let
be the sets of condition attribute subsets corresponding to
the prime implicants of F
B
ʲ
,
and
, F
P
ʲ
, and F
UN
ʲ
P
P
B
ʲ
P
ʲ
, respectively. Moreover, let
,
and
P
UN
ʲ
be the sets of condition attribute subsets corresponding to the prime implicants
of F
B
ʲ
, F
P
ʲ
, and F
UN
ʲ
, respectively. Then, we have the following implications:
B
for some B
∈
P
B
ʲ
•
If A is a B-reduct with
ʲ
then A
∈{
B
ↆ
C
|
B
ↇ
and
B
for any B
∈
P
B
B
Ↄ
ʲ
}
,
B
for some B
∈
P
P
ʲ
•
If A is a P-reduct with
ʲ
then A
∈{
B
ↆ
C
|
B
ↇ
and
B
for any B
∈
P
P
B
Ↄ
ʲ
}
,
P
B
for some B
∈
UN
ʲ
•
If A is a UN-reduct with
ʲ
then A
∈{
B
ↆ
C
|
B
ↇ
and
B
for any B
∈
P
UN
B
Ↄ
ʲ
}
.
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