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is the set of objects which have the same values as
u
for all attributes in
A
.We
denote the set of all equivalence classes with respect to
R
A
by
U
R
A
(
u
)
/
R
A
={
R
A
(
)
|
u
u
. Every equivalence class with respect to the decision attribute
d
is called a
decision class. For each value of the decision attribute
i
∈
U
}
∈
V
d
, we define the corre-
sponding decision class
X
i
={
u
∈
U
|
d
(
u
)
=
i
}
. Clearly,
X
={
X
1
,
X
2
,...,
X
p
}
forms a partition of
U
.
Example 2
Remember
be an
attribute subset. The discernibility relation
R
A
is described as the following matrix.
Symbol
D =
(
U
,
C
∪{
d
}
,
{
V
a
}
)
in Table
7.1
.Let
A
={
Pr
,
Ma
}
∗
indicates that the corresponding object pair
u
i
and
u
j
is in the discernibility
relation, i.e.,
(
u
i
,
u
j
)
∈
R
A
.
u
1
u
2
u
3
u
4
u
5
u
6
u
7
u
1
∗
u
2
∗∗ ∗∗
u
3
∗∗ ∗∗
u
4
∗
u
5
∗∗ ∗∗
u
6
∗∗ ∗∗
∗
From the matrix, we can easily see that the equivalence classes by
R
A
form a
partition of
U
, namely,
U
u
7
/
R
A
={{
u
1
}
,
{
u
4
}
,
{
u
7
}
,
{
u
2
,
u
3
,
u
5
,
u
6
}}
.
The decision classes of the decision table
D
are obtained as
X
unacc
={
u
1
,
u
2
}
,
X
acc
={
u
3
,
u
4
,
u
5
}
,
X
good
={
u
6
,
u
7
}
.
7.2.2 Rough Set Models
Let
A
be a subset of the attribute set
AT
and
X
be a subset of the object set
U
.
When
X
can be represented by a union of elements in
U
R
A
, we can say that the
classification by
X
is consistent with the information of
A
. Such subsets of objects are
called definable sets with respect to
A
. On the other hand, considering an object subset
X
which cannot be represented by any union of elements in
U
/
R
A
, the classification
of
X
is inconsistent with
A
. The classical Rough Set Model (RSM) [
35
,
36
,
39
]
deals with the inconsistency by two operators for object sets, called lower and upper
approximations. For
A
/
ↆ
AT
and
X
ↆ
U
, the lower approximation LA
A
(
X
)
and
the upper approximation UA
A
(
X
)
of
X
with respect to
A
is defined by:
LA
A
(
X
)
={
u
∈
U
|
R
A
(
u
)
ↆ
X
}
,
UA
A
(
X
)
={
u
∈
U
|
R
A
(
u
)
∩
X
=∅}
.
The difference between UA
A
(
X
)
and LA
A
(
X
)
is called the boundary of
X
with
respect to
A
, which is defined by:
BN
A
(
)
=
UA
A
(
)
\
LA
A
(
).
X
X
X
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