Question

Use the method from Exercise \(26\) to simplify the product-of-sums expansion \((x + y + z)(x + y + \bar z)(x + \bar y + \bar z)(x + \bar y + z)(\bar x + y + z)\).

Short answer

Simplified product \({\bf{x(y + z)}}\)

Step-by-step solution

Step 1:Definition

The product of the expression sumresults from the fact that two or more sums (OR's) are added (AND'ed) together. 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 product-of-sum

Product-of-sums expansions\((x + y + z)(x + y + \bar z)(x + \bar y + \bar z)(x + \bar y + z)(\bar x + y + z)\)

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

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

Place values in the cell

\(x + y + z\): Place a \(0\) in the cell in row \(x\) and column \({\bf{y + z}}\)

\(x + y + \bar z\): Place a \(0\) in the cell in row \(x\) and column \({\bf{y + \bar z}}\)

\(x + \bar y + \bar z\): Place a \(0\) in the cell in row \(x\) and column \({\bf{\bar y + \bar z}}\)

\(x + \bar y + z\): Place a \(0\) in the cell in row \(x\) and column \({\bf{\bar y + z}}\)

\(\bar x + y + z\): Place a \(0\) in the cell in row \(\bar x\) and column \({\bf{y + z}}\)

The row \(x\)contains only \(0\)'s, thus the largest block in the \(K{\bf{ - }}\)map is \(x\) (as there are empty cells). \(\bar x(y + z)\) was the only factor not included in the largest block: However \(\bar x(y + z)\) is included in the block \({\bf{y + z}}\) (as the column \({\bf{y + z}}\) contains only \(0\)'s ) and thus \(\bar x(y + z)\) is included in the block \({\bf{y + z}}\). The simplified product is then the product of the previous two blocks:

Therefore, this is the simplified product \({\bf{x(y + z)}}\).