Question

Find the sum-of-products expansions represented by each of these \(K{\bf{ - }}\)maps.

\(({\bf{a)}}\)

\({\bf{(b)}}\)

\({\bf{(c)}}\)

Short answer

\(({\bf{a)}}\)The sum-of-products expansion is then the sum of these three terms:\({\bf{xy + \bar xy + \bar x\bar y}}\).

\({\bf{(b)}}\)The sum-of-products expansion is then the sum of these three terms:\({\bf{xy + x\bar y}}\).

\({\bf{(c)}}\) The sum-of-products expansion is then the sum of these three terms:\({\bf{xy + x\bar y + \bar xy + \bar x\bar y}}\).

Step-by-step solution

Step 1:Definition

To reduce the number of terms in a Boolean expression representing a circuit, it is necessary to find terms to combine. There is a graphical method, called a Karnaugh map or K-map, for finding terms to combine for Boolean functions involving a relatively small number of variables. You will first illustrate how K-maps are used to simplify expansions of Boolean functions in two variables. You will continue by showing how K-maps can be used to minimize Boolean functions in three variables and then in four variables. Then you will describe the concepts that can be used to extend K-maps to minimize Boolean functions in more than four variables.

\(K{\bf{ - }}\)map for a function in two variables is basically a table with two columns \(y\) and \(\bar y\)and two rows \(x\) and \(\bar x\).

Finding the sum-of-products

(a)

Since the table contains a \(1\) in the row \(x\) and in the column \(y\), one of the terms in the sum-of-products expansion is \({\bf{xy}}\).

Since the table contains a \(1\) in the row \(x\) and in the column \(y\), one of the terms in the sum-of-products expansion is \({\bf{\bar xy}}\).

Since the table contains a \(1\) in the row \(x\) and in the column \(\bar y\), one of the terms in the sum-of-products expansion is \({\bf{\bar x\bar y}}\).

The sum-of-products expansion is then the sum of these three terms:

\({\bf{xy + \bar xy + \bar x\bar y}}\).

Finding the sum-of-products

(b)

\(K{\bf{ - }}\)map for a function in two variables is basically a table with two columns \(y\) and \(\bar y\)and two rows \(x\) and \(\bar x\).

Since the table contains a \(1\) in the row \(x\) and in the column \(y\), one of the terms in the sum-of-products expansion is \({\bf{xy}}\).

Since the table contains a \(1\) in the row \(x\) and in the column \(\bar y\), one of the terms in the sum-of-products expansion is \({\bf{\bar xy}}\).

The sum-of-products expansion is then the sum of these two terms:

\({\bf{xy + x\bar y}}\).

Finding the sum-of-products

(c)

\(K{\bf{ - }}\)map for a function in two variables is basically a table with two columns \(y\) and \(\bar y\)and two rows \(x\) and \(\bar x\).

Since the table contains a \(1\) in the row \(x\) and in the column \(y\), one of the terms in the sum-of-products expansion is \({\bf{xy}}\).

Since the table contains a \(1\) in the row \(x\) and in the column \(\bar y\), one of the terms in the sum-of-products expansion is \({\bf{x\bar y}}\).

Since the table contains a \(1\) in the row \(x\) and in the column \(y\), one of the terms in the sum-of-products expansion is \({\bf{\bar xy}}\).

Since the table contains a \(1\) in the row \(x\) and in the column \(\bar y\), one of the terms in the sum-of-products expansion is \({\bf{\bar x\bar y}}\).

The sum-of-products expansion is then the sum of these four terms:

\({\bf{xy + x\bar y + \bar xy + \bar x\bar y}}\).