Question

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

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

Short answer

\(\left( {\bf{a}} \right)\)The K-map of the sum-of-products is

\(\left( {\bf{b}} \right)\) The K-map of the sum-of-products is

\(\left( {\bf{c}} \right)\) The K-map of the sum-of-products is

Step-by-step solution

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. It will first illustrate how K-maps are used to simplify expansions of Boolean functions in two variables. It will continue by showing how K-maps can be used to minimize Boolean functions in three variables and then in four variables. Then it will describe the concepts that can be used to extend K-maps to minimize Boolean functions in more than four variables.

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 \({\bf{\bar x}}\).

Placing the value in the cell

(a)

Place a \(1\) in the cell corresponding to each term in the given sum \({\bf{x}}\overline {{\bf{yz}}} \).

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

Placing the value in the cell

(b)

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

It places a \(1\) in the cell corresponding to each term in the given sum \({\bf{\bar xyz + \bar x\bar y\bar z}}\).

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

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

Placing the value in the cell

(c)

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 \({\bf{\bar x}}\).

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

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

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

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

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