Consider the information system shown in Table 
. a and b are condition attributes, while d is
the decision attribute.
There is a total of 100 objects. In most cases, the value of d
follows from the value of a. However, the set is
not deterministic, as can be seen from the differing decision
values for the same condition values in classes and .
Attribute b seems to be irrelevant with respect to the value of d.
| a | b | d | # of objects | |
| 1 | 1 | 1 | 21 | |
| 1 | 1 | 0 | 2 | |
| 1 | 0 | 1 | 25 | |
| 0 | 1 | 0 | 26 | |
| 0 | 0 | 0 | 24 | |
| 0 | 0 | 1 | 2 |
Minimal relative reducts for this system are
As usual, C is the set of condition attributes. Deterministic rules can be derived from the reducts for the classes and . These are
As stated above, classes and are indeterministic. Default rules can be found using the reduct, though. These are:
From ¯:, 21/23 cases, 91.3%
, 2/23 cases, 8.7%
From :
, 24/26 cases, 92.3%
, 2/26 cases, 7.7%
Now, a new object is introduced with a=0 and b=2. The value of d is not given, and in the generated rule set there is no rule matching this object, yet. However, if attribute b is ignored, and this new data set is used when generating rules, additional rules would be created:
, 46/48 cases, 95.8%, 2/48 cases, 4.2%
, 50/52 cases, 96.2%
, 2/52 cases, 3.8%
It can be seen that both of the last two rules are applicable, but the rule is superior, with 96.2% validity compared to 3.8% validity for the second alternative. The most likely correct classification for the new object would therefore be to say that d=0. A number of additional rules to the ones listed here could be found using the algorithm. These would be rules for when only a is cut away, and rules for when both the attributes a and b are removed.
As fewer and fewer condition attributes are taken into consideration when generating rules, there will be formed a smaller amount of classes over the set. In other words, classes are so to speak ``melted'' together as attributes are removed. Above, the classes and were combined as a result of removing the attribute b. Similarly, and were combined. In the extreme case, no condition attributes are considered, and the probability of the decision class is its distribution. It is not necessary to produce rules for all possible subsets of condition attributes, as will be shown later in this chapter.