![](https://csdnimg.cn/release/download_crawler_static/15136313/bg4.jpg)
U − DIS
D
, then x
i
and x
j
are indistinguishable by D. at is,
[x
i
]
ind(D)
� [x
j
]
ind(D)
.
us, U × U − DIS
P
⊇U × U − DIS
C
is equivalent to any
(x
i
, x
j
) ∈ U × U, if [x
i
]
ind(C)
� [x
j
]
ind(C)
, then [x
i
]
ind(D)
�
[x
j
]
ind(D)
. at is, ind
C
⊆ind
D
. erefore, (2) ⟺ (3).
Similarly, it can be proved that (1) ⟺ (2).
□
Theorem 2. Let S � (U, C ∪ D) be a decision system, P ⊆C
and P ≠∅. DIS
P
has the following properties.
(1) If P
1
⊆P
2
⊆C, then DIS
P
1
⊆DIS
P
2
D ⊆DIS
C
(2) DIS
P
� ∪
a
k
∈P
DIS
a
k
Proof. e proof of eorem 2 is similar to that of eorem
2 in [27].
□
Theorem 3. Let S � (U, C ∪ D) be a decision system, P ⊆C
and P ≠∅, then |M
∗
P
| � |DIS
P
∩DIS
D
|.
Proof. According to Definition 8, for any a
k
∈ P, m
∗
k
�
DIS
a
k
∩DIS
D
, and |M
∗
P
| � | ∪m
k
∗
|. us, according to
property 2 in eorem 2, |M
∗
P
| � |DIS
P
∩DIS
D
|. □
Theorem 4. Let S � (U, C ∪D) be a decision system; then,
consistent reduction with distinguishability must exist.
Theorem 5. Let S � (U, C ∪ D) be a decision system. In
decision system S, consistent set and consistent attribute set
with distinguishability are equivalent.
Theorem 6. Let S � (U, C ∪ D) be a decision system. In
decision system S, attribute reduction and consistent re-
duction with distinguishability are equivalent.
Theorem 7. Let S � (U, C ∪ D) be a decision system, P ⊆C
and P ≠∅. e following propositions are equivalent.
(1) P is a consistent attribute set with distinguishability
(2) DIS
P
∩DIS
D
⊇DIS
C
∩DIS
D
(3) P ≡ C
Theorem 8. Let S � (U, C ∪ D) be a decision system, P ⊆C
and P ≠∅. e following propositions are equivalent.
(1) P is a consistent attribute set with distinguishability
(2) DIS
P
∩DIS
D
⊇DIS
C
∩DIS
D
, and ∀Q ⊂ P, DIS
Q
∩
DIS
D
≠DIS
C
∩DIS
D
(3) P ≡ C, and ∀Q ⊂ P, Q ≡ C is not true
The proof of above theorems is similar to the proof of
Theorems 11 to 15 in [29] (not elaborated here).
4. Decision Dependency Degree Based on
Distinguishable Relation
Based on the distinguishable relation, Definition 10 and
Definition 11 are proposed as follows. Definition 10
“Dependency Based on Distinguishable Relation” is used to
determine whether there is a dependency relationship be-
tween the attribute sets P
1
and P
2
. And Definition 11
“Dependency Degree based on Distinguishable Relation” is
used to assess the extent of the dependency or replacement
relationship that exists between P
1
and P
2
.
Definition 10 (dependency based on distinguishable
relation). Let S � (U, C ∪D) be a decision system,
P
1
, P
2
⊆C and P
1
, P
2
≠∅. en, the definition “dependency
based on distinguishable relation” is defined as follows:
(1) Attribute set P
2
depends on attribute set P
1
based on
distinguishable relation iff DIS
P
2
∩DIS
D
⊆DIS
P
1
∩DIS
D
. is decision dependency can be denoted as
P
1
⟹P
2
.
(2) Attribute set P
2
is equivalent to attribute set P
1
based
on distinguishable relation iff P
1
⟹P
2
and
P
2
⟹P
1
. is equivalent relationship can be
denoted as P
1
≡ P
2
. It is obvious that P
1
≡ P
2
, iff
DIS
P
2
∩DIS
D
⊆DIS
p
1
∩DIS
D
.
Definition 11 (distinguishable relation-based dependency
degree). Let S � (U, C ∪D) be a decision system, P
1
, P
2
⊆C
and P
1
, P
2
≠∅; if decision dependency degree k(0 ≤k ≤ 1) is
denoted as
k
p
2
P
1
�
1 − DIS
P
2
∩DIS
D
DIS
P
1
∩DIS
D
, (7)
then the extent to which P
2
depends on attribute set P
1
based on distinguishable relation is k(0 ≤k ≤1), called as
P
2
. k depends on P
1
(P
2
depends on P
1
in k decision
dependency degree), or P
1
can be substituted by P
2
in 1 − k
decision substitution degree, which is denoted as
P
1
k
⟹P
2
.
(1) If k � 0, named P
2
does not depend on P
1
based on
distinguishable relation, or P
1
can be completely
replaced (substituted) by P
2
(2) If 0 <k <1, named P
2
partly depends on P
1
based on
distinguishable relation, or P
1
can be partly replaced
(substituted) by P
2
(3) If k � 1, named P
2
completely depends on P
1
based
on distinguishable relation, or there does not exist
decision substitution relationship between P
1
and
P
2
Theorem 9. Let S � (U, C ∪D) be a decision system,
P
1
, P
2
⊆C, P
1
, P
2
≠∅, and P
1
⟹P
2
; then, the following
propositions are true:
(1) Decision dependency degree k
P
2
(P
1
) is 0 if |M
∗
P
1
|
� |M
∗
P
2
|
(2) Decision dependency degree k
P
2
(P
1
) is 1 if |M
∗
P
2
| � 0
or DIS
P
2
∩DIS
D
� ∅
4 Mathematical Problems in Engineering