Positive and Negative Classes

The classes, $$A,B,AB$$, are called positive classes because they contain all positive attributes. The classes $$\alpha ,\beta ,\alpha \beta $$ are called negative classes because they have negative attributes. The classes $$\alpha B$$ and $$A\beta $$ contain both positive and negative attributes, so they are called mixed or contrary classes.

If we have three attributes $$A,B,C$$ with their opponents or complements as $$\alpha ,\beta $$ and $$\gamma $$, then we can write the different class frequencies as in the table below:

In this table the positive classes are $$A,B,C,AB,AC,BC$$ and $$ABC$$ whereas the negative classes are$$\alpha ,\beta ,\gamma ,\alpha \beta ,\alpha \gamma ,\beta \gamma ,\alpha \beta \gamma $$. All other classes are mixed.

Order of Classes

The order of the class depends upon the number of attributes going into that class. If a certain class can give us information about only one attribute, it is called a class of order one.

The classes $$A,\alpha ,B$$ and $$\beta $$ are the classes of order one and the frequencies$$\left( A \right),\left( \alpha \right)$$, $$\left( B \right)$$ and $$\left( \beta \right)$$ are the frequencies of order one. The classes $$AB,A\beta $$, $$\alpha B$$ and $$\alpha \beta $$ are the classes of order two and the frequencies $$\left( {AB} \right),\left( {A\beta } \right),\left( {\alpha B} \right)$$ and $$\left( {\alpha \beta } \right)$$ are the frequencies of order two. The classes $$ABC,AB\gamma ,\alpha BC$$,$$\alpha B\gamma ,A\beta C$$, $$A\beta \gamma ,\alpha \beta C$$ and $$\alpha \beta \gamma $$ are the classes of order three and $$\left( {ABC} \right)$$, $$\left( {AB\gamma } \right)$$ … $$\left( {\alpha \beta \gamma } \right)$$ are the frequencies of order three. The sample size n does not contain any attributes and is therefore called the frequency of order zero.