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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
/
AT and X
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
)
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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