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Fig. 7.3
Strong-weak hierar-
chy of reducts in DRSM
strong
weak
7.4.3 Boolean Functions Representing Reducts
Because Boolean reasoning is a popular approach to enumerate all reducts of each
type in rough set literature, some authors already showed Boolean functions repre-
senting their own types of reducts [
49
,
52
]. On the other hand, the authors proposed
a unified formulation of Boolean functions for all types of reducts using the general-
ized decision function in [
31
]. We only discuss Boolean functions for L
≥
-, L
≤
- and
L-reducts, because U-reducts, LL
≥
-reducts, LL
≤
-reducts, and their equivalences can
be computed from L
≥
- and L
≤
-reducts or their Boolean functions.
We represent preserving conditions of reducts by those of the generalized decision
function.
Lemma 5
([
31
])
Let A be a subset of C . We have the following assertions.
Condition (
DL
≥
) is equivalent to:
•
l
A
(
u
)
=
l
C
(
u
)
for all u
∈
U
.
(DlG)
Condition (
DL
≤
) is equivalent to:
•
u
A
(
)
=
u
C
(
)
∈
.
u
u
for all u
U
(DuG)
•
Condition (DL) is equivalent to:
ʴ
A
(
u
)
=
ʴ
C
(
u
)
for all u
∈
U such that l
C
(
u
)
=
u
C
(
u
).
(DLG)
The next lemma is parallel to Lemma
2
of RSM. It also connects two notions:
“preserving” and “non-dominating”.
Lemma 6
([
31
])
Let u
∈
U . The following assertions are equivalent.
•
l
A
(
u
)
=
l
C
(
u
)
.
u
∈
u
)<
u
,
•∀
U
,(
l
C
(
l
C
(
u
)
⃒∃
a
∈
A
,(
u
)
∈
D
}
)
.
{
a
Moreover, the following assertions are also equivalent.
•
u
A
(
u
)
=
u
C
(
u
)
.
u
∈
u
)>
u
)
∈
•∀
U
,(
u
C
(
u
C
(
u
)
⃒∃
a
∈
A
,(
u
,
D
{
a
}
)
.
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