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:

 
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 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.