D. Chen et al. / International Journal of Approximate Reasoning 55 (2014) 908–923 911
Definition 2.1. (See [34,36].) Let U be a universe, and let C be a family of subsets of U . C is called a covering of U if none
of the subsets in C is empty and
C = U , and (U , C) is called a covering approximation space.
It is clear that a partition of U is
a covering of U , and hence, the concept of a covering is an extension of the concept
of a partition. As was noted in [6,7], a covering can be generated by set-valued attributes, missing-valued attributes and
real-valued attributes. Thus, for the appropriate types of datasets, coverings can be employed to substitute for attributes
when covering rough sets are employed as a mathematical tool to reduce superfluous attributes. Here, we demonstrate how
to induce a covering from attributes with several examples.
The dataset with set-valued attributes is an information system or decision system in which some objects have multiple
attribut
e values for a given attribute. Let U be a set of objects, and let V be a set of possible values of the attribute.
A multiple-valued mapping F
: U → V represents an imprecise description of the attributes’ values, and through the
multiple-valued mapping, we can definite the upper inverse of an attribute value F
∗
: V → U . Thus, a set-valued attribute
can induce a covering, and each set in the covering is the upper inverse of the image of an attribute value.
Example 2.2. (See [6].)
Let us consider the problem of evaluating credit card applicants. Suppose U ={x
1
,...,x
9
} is a
set of nine applicants, and let E
={education; salary} be a set of two attributes, where the values of “education” are
{higher; seeondary; primary} and the values of “salary” are {high; middle; low}. Suppose we have three specialists {A, B, C}
to evaluate the attribute values for these applicants. It is possible that their evaluation results of the same attribute may
not be the same. The results are listed below:
For the attribute “education”:
A: higher ={x
1
, x
4
, x
5
, x
7
}, secondary ={x
2
, x
8
}, primary ={x
3
, x
6
, x
9
};
B: higher ={x
1
, x
2
, x
4
, x
7
, x
8
}, secondary ={x
5
}, primary ={x
3
, x
6
, x
9
};
C: higher ={x
1
, x
4
, x
7
}, secondary ={x
2
, x
8
}, primary ={x
3
, x
5
, x
6
, x
9
}.
For the attribute “salary”:
A: higher ={x
1
, x
2
, x
3
}, middle ={x
4
, x
5
, x
6
, x
7
, x
8
}, low ={x
9
};
B: higher ={x
1
, x
2
, x
3
}, middle ={x
4
, x
5
, x
6
, x
7
}, low ={x
8
, x
9
};
C: higher ={x
1
, x
2
, x
3
}, middle ={x
4
, x
5
, x
6
, x
8
}, low ={x
7
, x
9
}.
The multiple-valued mappings for the attribute “education” are
F
e
(x
1
) ={higher}, F
e
(x
2
) ={higher, secondary},
F
e
(x
3
) ={primary}, F
e
(x
4
) ={higher}, F
e
(x
5
) ={higher, secondary, primary},
F
e
(x
6
) ={primary}, F
e
(x
7
) ={higher}, F
e
(x
8
) ={higher, secondary}, F
e
(x
9
) ={primary}.
The corresponding upper inverses are
F
∗
(higher) ={x
1
, x
2
, x
4
, x
5
, x
7
, x
8
}, F
∗
(secondary) ={x
2
, x
5
, x
8
}, F
∗
(primary) ={x
3
, x
5
, x
6
, x
9
}.
Thus, the covering C
1
induced by “education” is
C
1
=
{x
1
, x
2
, x
4
, x
5
, x
7
, x
8
}, {x
2
, x
5
, x
8
}, {x
3
, x
5
, x
6
, x
9
}
.
The multiple-valued mappings for the attribute “salary” are
F
s
(x
1
) ={high}, F
s
(x
2
) ={high}, F
s
(x
3
) ={high},
F
s
(x
4
) ={middle}, F
s
(x
5
) ={middle}, F
s
(x
6
) ={middle},
F
s
(x
7
) ={middle, low}, F
s
(x
8
) ={middle, low}, F
s
(x
9
) ={low}.
The corresponding upper inverses are
F
∗
(high) ={x
1
, x
2
, x
3
}, F
∗
(middle) ={x
4
, x
5
, x
6
, x
7
, x
8
}, F
∗
(low) ={x
7
, x
8
, x
9
}.
Thus, the covering C
2
induced by “education” is
C
2
=
{x
1
, x
2
, x
3
}, {x
4
, x
5
, x
6
, x
7
, x
8
}, {x
7
, x
8
, x
9
}
.
This example reflects the fact that set-valued attributes can induce a covering by using multiple-valued mappings.
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