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Therefore, discernibility functions
F
0
.
39
,
F
0
.
39
, and
F
UN
39
are obtained as:
0
.
F
0
.
39
(
˜
c
1
,
˜
c
2
,
˜
c
3
,
˜
c
4
)
=
m
ij
˜
c
∧
m
34
˜
c
=˜
c
1
∧˜
c
2
,
c
∈
c
∈
i
=
1
,
2
,
5
,
j
=
3
,
4
F
0
.
39
(
˜
c
1
,
˜
c
2
,
˜
c
3
,
˜
c
4
)
=
m
ij
˜
c
=
(
˜
c
1
)
∧
(
˜
c
2
∨˜
c
3
)
∧
(
˜
c
2
∨˜
c
4
)
c
∈
i
=
1
,
2
,
j
=
3
,
4
,
5
=
(
˜
c
1
∧˜
c
2
)
∨
(
˜
c
1
∧˜
c
3
∧˜
c
4
),
F
UN
0
39
(
˜
c
1
,
˜
c
2
,
˜
c
3
,
˜
c
4
)
=
m
ij
˜
c
=
(
˜
c
1
∨˜
c
3
)
∧
(
˜
c
2
∨˜
c
3
)
∧
(
˜
c
2
∨˜
c
4
)
.
c
∈
i
=
1
,
2
,
3
,
4
,
j
=
5
=
(
˜
c
1
∧˜
c
2
)
∨
(
˜
c
2
∧˜
c
3
)
∨
(
˜
c
3
∧˜
c
4
).
Because
F
0
.
39
=
F
0
.
39
∧
F
0
.
39
=
(
c
1
∧˜
˜
c
2
∧˜
c
3
)
∨˜
c
1
∧˜
c
2
∧˜
c
4
)
, the candidates
of B-reducts are,
{
.
We can see that all of those satisfy (
VPB1
), hence,
c
1
,
c
2
}
,
{
c
1
,
c
2
,
c
3
}
,
{
c
1
,
c
2
,
c
4
}
{
c
1
,
c
2
}
is the unique B-reduct.
Because
F
0
.
39
=
F
0
.
39
=
(
˜
c
1
∧˜
c
2
∧˜
c
3
)
∨
(
˜
c
1
∧˜
c
2
∧˜
c
4
)
∨
(
˜
c
1
∧˜
c
3
∧˜
c
4
)
,the
candidates of P-reducts are,
{
.
Also, in that case, all candidates satisfy (
VPP1
), hence,
c
1
,
c
2
}
,
{
c
1
,
c
2
,
c
3
}
,
{
c
1
,
c
2
,
c
4
}
,
{
c
1
,
c
3
,
c
4
}
{
c
1
,
c
2
}
and
{
c
1
,
c
3
,
c
4
}
are
P-reducts. Similarly, the candidates of UN-reducts are,
{
c
1
,
c
2
}
,
{
c
2
,
c
3
}
,
{
c
3
,
c
4
}
,
{
c
1
,
c
2
,
c
4
}
,
{
c
1
,
c
3
,
c
4
}
,
and all candidates satisfy (
VPUN1
), hence,
{
c
1
,
c
2
}
,
{
c
2
,
c
3
}
, and
{
c
3
,
c
4
}
are
UN-reducts.
All reducts are arranged in Table
7.6
. We can observe that several kinds of reducts
are different. In this example, each L-reduct is also an LUN-reduct and vice versa.
Such an equivalence holds in this example but not always.
to preserve all struc-
tures. Additionally,
c
1
and
c
2
appear in many other reducts. Whereas, we would
select U-reduct
In this example, we would select
{
c
1
,
c
2
,
c
3
}
or
{
c
1
,
c
2
,
c
4
}
{
c
2
,
c
3
}
to reduce the size of the reduct.
Table 7.6
All obtained
reducts with
Type
Reducts
ʲ
=
0
.
39 in
L-reduct
{
c
1
,
c
2
,
c
3
}
,
{
c
1
,
c
2
,
c
4
}
,
{
c
1
,
c
3
,
c
4
}
Tabl e
7.3
U-reduct
{
c
2
,
c
3
}
,
{
c
1
,
c
2
,
c
4
}
LU-reduct
{
c
1
,
c
2
,
c
3
}
,
{
c
1
,
c
2
,
c
4
}
B-reduct
{
c
1
,
c
2
}
P-reduct
{
c
1
,
c
2
}
,
{
c
1
,
c
3
,
c
4
}
UN-reduct
{
c
1
,
c
2
}
,
{
c
2
,
c
3
}
,
{
c
3
,
c
4
}
LUN-reduct
{
c
1
,
c
2
,
c
3
}
,
{
c
1
,
c
2
,
c
4
}
,
{
c
1
,
c
3
,
c
4
}
BUN-reduct
{
c
1
,
c
2
}
PUN-reduct
{
c
1
,
c
2
}
,
{
c
1
,
c
3
,
c
4
}
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