Question

Draw the \(K{\bf{ - }}\)maps of these sum-of-products expansions in two variables.

\(\begin{array}{l}{\bf{a)x\bar y}}\\{\bf{b)xy + \bar x\bar y}}\\{\bf{c)xy + x\bar y + \bar xy + \bar x\bar y}}\end{array}\)

Short answer

\((a)\) The K-map of the given expansion \({\bf{x\bar y}}\) is

\((b)\) The K-map of the given expansion \({\bf{xy + \bar x\bar y}}\) is

\((c)\) The K-map of the given expansion \({\bf{xy + x\bar y + \bar xy + \bar x\bar y}}\) is

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.

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

Mapping of sum-of-products

(a)

It places a \(1\) in the cell corresponding to \({\bf{x\bar y}}\) (which is the in the cell in the row \(x\) and in the column \({\bf{\bar y}}\).

Mapping of sum-of-products

(b)

Place a\(1\) in the cell corresponding to each term in the given sum \({\bf{xy + \bar x\bar y}}\).

\({\bf{xy}}\): place a \(1\) in the cell corresponding to row \(x\) and column \({\bf{y}}\).

\({\bf{\bar x\bar y}}\):place a \(1\) in the cell corresponding to row \({\bf{\bar x}}\) and column \({\bf{\bar y}}\).

Mapping of sum-of-products

(c)

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

Place a \(1\) in the cell corresponding to each term in the given sum \({\bf{xy + x\bar y + \bar xy + \bar x\bar y}}\).

\({\bf{xy}}\): place a \(1\) in the cell corresponding to row \(x\) and column \({\bf{y}}\).

\({\bf{x\bar y}}\): place a \(1\) in the cell corresponding to row \(x\) and column \({\bf{\bar y}}\).

\({\bf{\bar xy}}\): place a \(1\) in the cell corresponding to row \({\bf{\bar x}}\) and column \({\bf{y}}\).

\({\bf{\bar x\bar y}}\): place a \(1\) in the cell corresponding to row \({\bf{\bar x}}\) and column \({\bf{\bar y}}\).