Question

Explain how \({\bf{K}}\)-maps can be used to simplify product-of-sums expansions in three variables. (Hint: Mark with a\(0\) all the max-terms in an expansion and combine blocks of maxterms.)

Short answer

Place a \(0\) in the table for every given maxterm (sum) in the expansion in the corresponding cell.

Combine squares to produce larger blocks

Simplification isthe product of the maxterms corresponding to the largest blocks and such that each term of the initial expansion occurs in one of the maxterms.

Step-by-step solution

Step 1:Definition

A product sum expansion, or disjunctive normal form, of a Boolean function is the function written as a sum of minterms. The product of the expression sum results from the fact that two or more sums (OR) are added (AND). That is the outputs from two or more OR gates are connected to the input of an AND gate so that they are effectively AND'ed together to create the final (OR AND) output.

Using the sum-of-products and products-of-sum

Sum-of-products expansions:

A \({\bf{K}}\)-map for a function in three variables is a table with four columns \({\bf{yz, y\bar z,\bar y\bar z}}\) and \({\bf{\bar yz}}\); which contains all possible combinations of \({\bf{y}}\) and \({\bf{z}}\)and two rows \({\bf{x}}\) and \({\bf{\bar x}}\).

Product-of-sums expansions:

Use the same \({\bf{K}}\)-map as for the sum-of-products expansions, except that we replace the products in the column titles by sums thus \({\bf{yz}}\) is replaced by \({\bf{y + z,y\bar z}}\) is replaced by \({\bf{y + \bar z,\bar y\bar z}}\) is replaced by \({\bf{\bar y + \bar z,\bar yz}}\) is replaced by \({\bf{\bar y + z}}\)

Placing values in the table

Place a \({\bf{0}}\) in the table for every given maxterm (sum) in the expansion in the corresponding cell.

For example, if the expansion contains \({\bf{x + y + z}}\), then we place a \({\bf{0}}\) in the cell in the row \({\bf{x}}\) and column \({\bf{yz}}\).

Combine squares to produce larger blocks.

The product of the maxterms corresponding to the largest blocks and such that each term of the initial development is found in one of the maxterms.