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By the duality of lower and upper approximations, the boundaries of the pair of
complementary unions are the same,
X
i
X
i
−
1
).
BN
A
(
)
=
BN
A
(
(7.25)
Lower and upper approximations can be expressed by boundaries. That is useful
for investigating relations between different types of reducts:
X
i
X
i
X
i
X
i
X
i
X
i
UA
A
(
)
=
BN
A
(
)
∪
,
UA
A
(
)
=
BN
A
(
)
∪
,
(7.26)
X
i
X
i
X
i
X
i
X
i
X
i
LA
A
(
)
=
\
BN
A
(
),
LA
A
(
)
=
\
BN
A
(
).
(7.27)
,
ↆ
,
∈
Let
A
B
C
and
i
j
V
d
. Then, we have the following monotonicity proper-
ties:
X
j
)
ↆ
X
i
X
j
)
ↆ
X
i
j
≥
i
⃒
LA
A
(
LA
A
(
),
UA
A
(
UA
A
(
),
(7.28)
X
j
)
ↆ
X
i
X
j
)
ↆ
X
i
j
≤
i
⃒
LA
A
(
LA
A
(
),
UA
A
(
UA
A
(
),
(7.29)
X
i
X
i
X
i
X
i
B
ↆ
A
⃒
LA
B
(
)
ↆ
LA
A
(
),
LA
B
(
)
ↆ
LA
A
(
),
(7.30)
X
i
X
i
X
i
X
i
B
ↆ
A
⃒
UA
B
(
)
ↇ
UA
A
(
),
UA
B
(
)
ↇ
UA
A
(
).
(7.31)
Those are important for defining and enumerating reducts.
Furthermore, the authors proposed lower and upper approximations and boundary
regions of decision classes [
31
]. For
A
V
d
, lower and upper approxi-
mations of
X
i
and the boundary region of
X
i
are defined by:
ↆ
C
and
i
∈
X
i
X
i
LA
A
(
X
i
)
=
LA
A
(
)
∩
LA
A
(
),
X
i
X
i
UA
A
(
X
i
)
=
UA
A
(
)
∩
UA
A
(
),
BN
A
(
X
i
)
=
UA
A
(
X
i
)
\
LA
A
(
X
i
).
X
i
X
i
This definition is an analogy to
X
i
=
∩
.
V
d
. The upper approximations of
X
i
and
X
i
Let
A
ↆ
C
and
i
∈
are represented
by upper approximations of decision classes:
X
i
UA
A
(
)
=
UA
A
(
X
j
),
(7.32)
j
≥
i
X
i
UA
A
(
)
=
UA
A
(
X
j
).
(7.33)
j
≤
i
The boundary of
X
i
is the union of the boundaries of
X
i
and
X
i
,
X
i
X
i
BN
A
(
X
i
)
=
BN
A
(
)
∪
BN
A
(
).
(7.34)
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