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From the table, we obtain
F
0
.
39
and
F
0
.
39
as follows:
F
0
.
39
(
˜
c
1
,
˜
c
2
,
˜
c
3
,
˜
c
4
)
=
m
ij
˜
c
∧
m
12
˜
c
c
∈
c
∈
i
=
1
,
2
,
j
=
3
,
4
,
5
=
(
˜
c
1
)
∧
(
˜
c
2
∨˜
c
3
)
∧
(
˜
c
2
∨˜
c
4
)
∧
(
˜
c
3
∨˜
c
4
)
=
(
˜
c
1
∧˜
c
2
∧˜
c
3
)
∨
(
˜
c
1
∧˜
c
2
∧˜
c
4
)
∨
(
˜
c
1
∧˜
c
3
∧˜
c
4
),
F
0
.
39
(
˜
c
1
,
˜
c
2
,
˜
c
3
,
˜
c
4
)
=
m
ij
˜
c
c
∈
(
i
,
j
)
∈{
(
k
,
l
)
|
k
=
l
}\{
(
1
,
3
),(
3
,
1
)
}
=
(
˜
c
2
)
∧
(
˜
c
1
∨˜
c
3
)
∧
(
˜
c
3
∨˜
c
4
)
=
(
˜
c
2
∧˜
c
3
)
∨
(
˜
c
1
∧˜
c
2
∧˜
c
4
).
We obtain
F
LU
0
39
as:
.
F
LU
0
F
L
F
U
39
(
˜
c
1
,
˜
c
2
,
˜
c
3
,
˜
c
4
)
=
(
˜
c
1
,
˜
c
2
,
˜
c
3
,
˜
c
4
)
∧
(
˜
c
1
,
˜
c
2
,
˜
c
3
,
˜
c
4
)
.
=
(
˜
c
1
)
∧
(
˜
c
2
)
∧
(
˜
c
3
∨˜
c
4
)
=
(
˜
c
1
∧˜
c
2
∧˜
c
3
)
∨
(
˜
c
1
∧˜
c
2
∧˜
c
4
).
Therefore, L-reducts are obtained as
{
c
1
,
c
2
,
c
3
}
,
{
c
1
,
c
2
,
c
4
}
and
{
c
1
,
c
3
,
c
4
}
.
U-reducts are obtained as
{
c
2
,
c
3
}
and
{
c
1
,
c
2
,
c
4
}
. LU-reducts are obtained as
{
is not an L-reduct but a U-reduct.
This is very different from the relation between L- and U-reducts in the classical
RSM, i.e., in the classical RSM, a U-reduct includes an L-reduct but an L-reduct
never includes a U-reduct.
Now let us discuss B-, P-, UN-reducts with
c
1
,
c
2
,
c
3
}
and
{
c
1
,
c
2
,
c
4
}
. Note that
{
c
2
,
c
3
}
39. We can obtain only approx-
imations of those reducts. To this end, let us get discernibility functions
F
0
.
39
,
F
0
.
39
,
and
F
UN
0
ʲ
=
0
.
B
0
39
. For B-reducts, considering the second condition of
ʔ
39
, check each pair
.
.
0
.
39
0
.
39
P
i
and
P
j
such that
(˅
\
ʻ)
(
P
i
)
=∅
and
(˅
\
ʻ)
(
P
j
)
=∅
.
C
C
0
.
39
0
.
39
0
.
39
0
.
39
(˅
\
ʻ)
m
12
(
P
1
)
=
(˅
\
ʻ)
m
12
(
P
2
)
=∅
,(˅
\
ʻ)
m
15
(
P
1
)
=
(˅
\
ʻ)
m
15
(
P
5
)
=∅
,
C
\
C
\
C
\
C
\
0
.
39
0
.
39
(˅
\
ʻ)
m
25
(
P
2
)
=
(˅
\
ʻ)
m
25
(
P
5
)
=∅
.
C
\
C
\
0
.
39
0
.
39
For P-reducts, check each pair such that
ʻ
(
P
i
)
=∅
and
ʻ
(
P
j
)
=∅
.
C
C
0
.
39
0
.
39
ʻ
m
12
(
P
1
)
=
ʻ
m
12
(
P
2
)
={
medium
}
.
C
\
C
\
0
.
39
0
.
39
Finally, for UN-reducts, check each pair such that
˅
(
P
i
)
=∅
and
˅
(
P
j
)
=∅
.
C
C
0
.
39
0
.
39
0
.
39
0
.
39
˅
m
12
(
P
1
)
=
˅
m
12
(
P
2
)
={
}
,˅
m
13
(
P
1
)
=
˅
m
13
(
P
3
)
={
}
,
medium
good
C
\
C
\
C
\
C
\
0
.
39
0
.
39
0
.
39
0
.
39
˅
m
14
(
P
1
)
=
˅
m
14
(
P
4
)
={
good
}
,˅
m
23
(
P
2
)
=
˅
m
23
(
P
3
)
={
medium
}
,
C
\
C
\
C
\
C
\
0
.
39
0
.
39
0
.
39
0
.
39
˅
m
24
(
P
2
)
=
˅
m
24
(
P
4
)
={
}
,˅
m
34
(
P
3
)
=
˅
m
34
(
P
4
)
={
}
.
medium
good
C
\
C
\
C
\
C
\
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