Maxterms

A maxterm is a Boolean expression composed of a logical OR combination of Boolean variables $ x_i $, designed to render the function output false (y=0). Each variable within the expression can be in its direct form $ x_1 $ if it equals 0, or in its complemented form $ \overline{x_1} $ if it equals 1.

Maxterms outline specific scenarios where a Boolean function's output is false.

These expressions are crucial for specifying when certain circuits or logical functions will result in a "0" (false).

Example

Consider a Boolean function with three input variables: $ x_1 $, $ x_2 $, and $ x_3 $.

$$ y = f(x_1, x_2, x_3) $$

Here, $ x_1 , x_2 , x_3 $ are the inputs to the function, and $ y $ denotes the output.

Assume the function has the following truth table:

$$ \begin{array}{|c|c|c|c|} \hline m & x_1 & x_2 & x_3 & y \\ \hline m_1 & 0 & 0 & 0 & 1 \\ m_2 & 0 & 0 & 1 & 0 \\ m_3 & 0 & 1 & 0 & 0 \\ m_4 & 0 & 1 & 1 & 1 \\ m_5 & 1 & 0 & 0 & 0 \\ m_6 & 1 & 0 & 1 & 1 \\ m_7 & 1 & 1 & 0 & 1 \\ m_8 & 1 & 1 & 1 & 0 \\ \hline \end{array} $$

To identify the maxterms of a Boolean function, examine the truth table rows where the output $ y $ is zero (0).

These rows are:

$$ \begin{array}{|c|c|c|c|c|} \hline m & x_1 & x_2 & x_3 & y \\ \hline m_2 & 0 & 0 & 1 & 0 \\ m_3 & 0 & 1 & 0 & 0 \\ m_5 & 1 & 0 & 0 & 0 \\ m_8 & 1 & 1 & 1 & 0 \\ \hline \end{array} $$

We then construct maxterms for these combinations:

For $ m_2 $ (0, 0, 1), the function output is false when $ x_1 = 0 $, $ x_2 = 0 $, and $ x_3 = 1 $. The maxterm is $ M_2 = x_1 + x_2 + \overline{x_3} $.
In maxterms, express the variable in its direct form $ x_i $ if it equals 0, and in its complemented form $ \overline{x_i} $ if it equals 1. Here, $ x_1 $ and $ x_2 $ are direct, while $ x_3 $ is complemented as $ \overline{x_3} $.
For $ m_3 $ (0, 1, 0), the output is false with $ x_1 = 0 $, $ x_2 = 1 $, and $ x_3 = 0 $. This maxterm is $ M_3 = x_1 + \overline{x_2} + x_3 $.
For $ m_5 $ (1, 0, 0), with $ x_1 = 1 $, $ x_2 = 0 $, and $ x_3 = 0 $, the corresponding maxterm is $ M_5 = \overline{x_1} + x_2 + x_3 $.
For $ m_8 $ (1, 1, 1), the output is false when all inputs are 1. The maxterm then is $ M_8 = \overline{x_1} + \overline{x_2} + \overline{x_3} $.

The defined maxterms are:

$$ M_2 = x_1 + x_2 + \overline{x_3} $$

$$ M_3 = x_1 + \overline{x_2} + x_3 $$

$$ M_5 = \overline{x_1} + x_2 + x_3 $$

$$ M_8 = \overline{x_1} + \overline{x_2} + \overline{x_3} $$

These maxterms pinpoint the exact conditions under which the Boolean function's output is false.

Combining all the maxterms, the result is the negation of the original function:

$$ \overline{y} = M_2 \cdot M_3 \cdot M_5 \cdot M_8 $$

This formulation ensures that the overall expression is only true when all the maxterms are true simultaneously, a condition that never occurs in the function's original setup, thereby making the overall expression false when the function is true, and vice versa.

This logical product of maxterms effectively creates a comprehensive negation of the original function.