Question

Explain how to construct the sum-of-products expansion of a Boolean function.

Short answer

Create a table of values for \({\bf{F}}\) or use identity laws.

Step-by-step solution

Definition

Sum-of-products expansion:

There are two methods to determine the sum-of-products expansion of a Boolean function.

First method Create a table that determines the value of \({\bf{F}}\) for every possible combination of the values of the variables.

For every row of the table that contains a \({\bf{1}}\) for \({\bf{F}}\), there is a product of the form \({{\bf{y}}_{\bf{1}}}{{\bf{y}}_{\bf{2}}}{\bf{ \ldots }}{{\bf{y}}_{\bf{n}}}\) in the sum-of-products expansion where \({{\bf{y}}_{\bf{i}}}{\bf{ = }}{{\bf{x}}_{\bf{i}}}\) if \({{\bf{x}}_{\bf{i}}}{\bf{ = 1}}\) and \({{\bf{y}}_{\bf{i}}}{\bf{ = }}{{\bf{\bar x}}_{\bf{i}}}\) if \({{\bf{x}}_{\bf{i}}}{\bf{ = 0}}\).

Using the sum-of-products expansion

The sum-of-products expansion is then the sum of all the products.

Second method Use identity laws to derive an equivalent expression for the function \({\bf{F}}\) that contains only sums of products.

Hence,it needs to create a table of values for \({\bf{F}}\) or use identity laws.