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is interpreted as the set of objects which are certainly classified to
X
in
view of
A
. While, UA
A
(
LA
A
(
X
)
)
X
is the set of objects which are possibly classified to
X
in
view of
A
.BN
A
(
is a set of objects whose membership to
X
is doubtful.
The approximations are a definable set with respect to
A
, where a definable set
with respect to
A
is a set defined by the union of elements in
U
X
)
/
R
A
:
LA
A
(
X
)
=
R
A
(
u
)
=
R
A
(
u
),
R
A
(
u
)
ↆ
X
u
∈
LA
A
(
X
)
UA
A
(
X
)
=
R
A
(
u
)
=
R
A
(
u
).
R
A
(
u
)
∩
X
=∅
u
∈
UA
A
(
X
)
The boundary is necessarily definable because
U
/
R
A
is the partition of
U
.
In fact, LA
A
(
X
)
and UA
A
(
X
)
are “lower” and “upper” approximations of
X
i
:
LA
A
(
X
)
ↆ
X
ↆ
UA
A
(
X
).
(7.1)
By the above inclusion relations and the definition of the boundary, it holds that
LA
A
(
X
)
=
X
\
BN
A
(
X
),
(7.2)
UA
A
(
X
)
=
X
∪
BN
A
(
X
).
(7.3)
ↂ
ↆ
For
B
A
AT
,wehave,
LA
B
(
X
)
ↆ
LA
A
(
X
)
and UA
B
(
X
)
ↇ
UA
A
(
X
).
(7.4)
When
B
is included in
A
, the approximations with respect to
B
are coarser that
those with respect to
A
. It means that dropping some attributes, i.e., information,
decline the accuracy of RSM.
So far, we have defined approximations of
X
from the lower and upper definable
sets. We can approximate the partition
X
and
U
X
by three definable sets. They
are called positive, boundary, and negative regions of
X
with respect to
A
, denoted
by POS
A
(
\
X
)
,BND
A
(
X
)
, and NEG
A
(
X
)
, respectively:
POS
A
(
X
)
=
{
E
∈
U
/
R
A
|
E
ↆ
X
}
,
BND
A
(
X
)
=
{
E
∈
U
/
R
A
|
E
∩
X
=∅
and
E
∩
U
\
X
=∅}
,
NEG
A
(
)
=
{
∈
/
R
A
|
ↆ
\
}
.
X
E
U
E
U
X
POS
A
(
X
)
is the union of elements in
U
/
R
A
which are completely included in
X
,
while NEG
A
(
X
)
is the union of elements in
U
/
R
A
which are completely excluded
from
X
.BND
A
(
X
)
is the union of the rest of elements in
U
/
R
A
.Clearly,POS
A
(
X
)
,
BND
A
(
X
)
, and NEG
A
(
X
)
form a partition of
U
. We can easily see the following
correspondence:
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