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Table 7.8
The decision table
in Table
7.7
with the
generalized decision function
Student Ma
Ph
Li
E
ʴ
=
l
C
,
u
C
C
u
1
Good Good Good Good
good, good
u
2
Good Good Med Med
med, good
u
3
Med
Good Med
Good
med, good
u
4
Bad
Med
Good Med
med, med
u
5
Med
Bad
Med
Bad
bad, med
u
6
Med
Bad
Bad
Med
bad, med
u
7
Bad
Bad
Bad
Bad
bad, bad
Non-domination matrices
M
are obtained as follows.
u
1
u
2
u
3
u
4
u
5
u
6
u
7
u
1
∅{
}{
,
}{
,
}
Li
Ma
Li
Ma
Ph
C
C
C
u
2
∅∅ {
}{
,
}{
,
}
Ma
Ma
Ph
Ma
Ph
C
C
u
3
∅∅ ∅ {
Ma
,
Ph
}{
Ph
}{
Ph
,
Li
}
C
u
4
∅{
Li
}{
Li
}
∅
{
Ph
,
Li
}{
Ph
,
Li
}{
Ph
,
Li
}
u
5
∅∅ ∅
{
Ma
}
∅
{
Li
}{
Ma
,
Li
}
u
6
∅∅ ∅
{
Ma
}
∅
∅
{
Ma
}
u
7
∅∅ ∅
∅
∅
∅
∅
For example, the entry corresponding to row
u
1
and column
u
3
on
M
contains
Ma and Li, because
u
3
is worse than
u
1
with respect to Ma and Li but not worse
with respect to Ph. Symbol
C
at some entries means {Ma, Ph, Li}. The rows with
symbol
∗
show objects
u
i
such that
l
C
(
u
i
)
=
u
C
(
u
i
)
.
The Boolean function
F
≥
is obtained from
M
as
F
≥
(
Ma
,
Ph
,
Li
=
Ph
∧
Li
)
=
m
ij
˜
c
∧
m
ij
˜
c
.
c
∈
c
∈
i
=
1
,
j
=
2
,
3
,...,
7
i
=
2
,
3
,
4
,
j
=
5
,
6
,
7
From the last equation,
F
≥
(
Ma
,
Ph
,
Li
true only when Ph
true and Li
true.
This implies that only {Ma, Ph, Li} and {Ph, Li} satisfy (DL
≥
) owing to Theorem
10
.
An L
≥
-reduct is a minimal set of condition attributes that satisfies (DL
≥
). Therefore,
{Ph, Li} is a unique L
≥
-reduct. Moreover, the L
≥
-reduct corresponds to a unique
prime implicant of
F
≥
, i.e., Ph
)
=
=
=
∧
Li.
Similarly, Boolean functions
F
≤
,
F
U
and
F
L
are
F
≤
(
Ma
Ph
Li
Ma
Ph
,
,
)
=
c
∧
c
=
∧
,
c
∈
m
ji
c
∈
m
ji
i
=
4
,
5
,
6
,
7
,
j
=
1
,
2
,
3
i
=
7
,
j
=
1
,
2
,...,
6
F
L
Ma
Ph
Li
(
,
,
)
=
c
∧
c
∧
c
c
∈
m
ij
c
∈
m
ij
c
∈
m
ji
i
=
1
,
j
=
2
,
3
,...,
7
i
=
4
,
j
=
5
,
6
,
7
i
=
4
,
j
=
1
,
2
,
3
Ma
Li
∧
c
=
∧
.
c
∈
m
ji
i
=
7
j
=
1
,
2
,...,
6
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