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, 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 below in table as shown.

 $A$ $\alpha$ $C$ $\gamma$ $C$ $\gamma$ Total $B$ $\left( {ABC} \right)$ $\left( {AB\gamma } \right)$ $\left( {\alpha BC} \right)$ $\left( {\alpha B\gamma } \right)$ $\left( B \right)$ $\beta$ $\left( {A\beta C} \right)$ $\left( {A\beta \gamma } \right)$ $\left( {\alpha \beta C} \right)$ $\left( {\alpha \beta \gamma } \right)$ $\left( \beta \right)$ $\left( {AC} \right)$ $\left( {A\gamma } \right)$ $\left( {\alpha C} \right)$ $\left( {\alpha \gamma } \right)$ Total $\left( A \right)$ $\left( \alpha \right)$ $n$

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 class of order one.

The classes $A,\alpha ,B$ and $\beta$ are classes of the 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 attribute and is therefore called frequency of order zero.