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9. Discovery of Positive and NegativeRules
239
9.3.5 Negative Rules
Before defining a negative rule, let us first introduce an exclusive rule, the
contrapositive of a negative rule [9.10]. An exclusive rule is defined as a rule
supported by all the positive examples, the coverage of which is equal to 1.0.
That is, an exclusive rule represents the necessity condition of a decision. It
is notable that the set supporting an exclusive rule corresponds to the upper
approximation of a target concept, which is introduced in rough sets [9.5].
Thus, an exclusive rule is represented as:
R
∨
j
[a
j
= v
k
],
R
(D)=1.0.
Figure 9.5 shows the Venn diagram of an exclusive rule. As shown in this
figure, the meaning of R is a superset of that of D. This diagram is exactly
equivalent to the classic proposition d
→
d
s.t.
R =
R.
In the preceding example, the exclusive rule of m.c.h. is:
[M1=yes]
→
m.c.h. κ =1.0,
From the viewpoint of propositional logic, an exclusive rule should be repre-
sented as:
d
∨
[nau = no]
→
→∨
j
[a
j
= v
k
],
because the condition of an exclusive rule corresponds to the necessity con-
dition of conclusion d. Thus, it is easy to see that a negative rule is defined
as the contrapositive of an exclusive rule:
∧
j
¬
d,
which means that if a case does not satisfy any attribute value pairs in the
condition of a negative rule, then we can exclude a decision d from candidates.
For example, the negative rule of m.c.h. is:
[a
j
= v
k
]
→¬
R
A
D
Fig. 9.5.
Venn diagram of exclusive rules.
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