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Chapter 12: Q15E (page 842)

Find the cells in a \(K{\bf{ - }}\)map for Boolean functions with five variables that correspond to each of these products.

\(\begin{array}{c}a){x_1}{x_2}{x_3}{x_4}\\b){{\bar x}_1}{x_3}{x_5}\\c){x_2}{x_4}\\d){{\bar x}_3}{{\bar x}_4}\\e){x_3}\\f){{\bar x}_5}\end{array}\)

Short Answer

Expert verified

\((a)\) A K-map for a function with five variables is

\((b)\) A K-map for a function with five variables is

\((c)\) A K-map for a function with five variables is

\((d)\) A K-map for a function with five variables is

\((e)\) A K-map for a function with five variables is

\((f)\) A K-map for a function with five variables is

Step by step solution

01

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.

02

Placing the values in the cells

A \(K{\bf{ - }}\)map for a function in five variables is a table with \(8\) columns \({x_3}{x_4}{x_5},{x_3}{x_4}{\bar x_5},{x_3}{\bar x_4}{\bar x_5},{x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{\bar x_5},{\bar x_3}{x_4}{\bar x_5}\) and \({\bar x_3}{x_4}{x_5}\); which contains all possible combinations of \({x_3},{x_4}\) and \({x_5}\) and four rows \({x_1}{x_2},{x_1}{\bar x_2},{\bar x_1}{\bar x_2}\) and \({\bar x_1}{x_2}\); which contains all possible combinations of \({x_1}\) and \({x_2}\).

Need to place a \(1\) in all cells corresponding with \({x_1}{x_2}{x_3}{x_4}\), which are the cells in the row \({x_1}{x_2}\) and in the columns \({x_3}{x_4}{x_5}/{x_3}{x_4}{\bar x_5}\)

03

Placing the values in the cells

A \(K{\bf{ - }}\)map for a function in five variables is a table with \(8\) columns \({x_3}{x_4}{x_5},{x_3}{x_4}{\bar x_5},{x_3}{\bar x_4}{\bar x_5},{x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{\bar x_5},{\bar x_3}{x_4}{\bar x_5}\) and \({\bar x_3}{x_4}{x_5}\); which contains all possible combinations of \({x_3},{x_4}\) and \({x_5}\) and four rows \({x_1}{x_2},{x_1}{\bar x_2},{\bar x_1}{\bar x_2}\) and \({\bar x_1}{x_2}\); which contains all possible combinations of \({x_1}\) and \({x_2}\).

Need to place a \(1\) in all cells corresponding with \({\bar x_1}{x_3}{x_5}\), which are the cells in the row \({\bar x_1}{x_2}/{\bar x_1}{\bar x_2}\) and in the columns \({x_3}{x_4}{x_5}/{x_3}{\bar x_4}{x_5}\)

04

Placing the values in the cells

\(K{\bf{ - }}\)map for a function in five variables is a table with \(8\) columns \({x_3}{x_4}{x_5},{x_3}{x_4}{\bar x_5},{x_3}{\bar x_4}{\bar x_5},{x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{\bar x_5},{\bar x_3}{x_4}{\bar x_5}\) and \({\bar x_3}{x_4}{x_5}\); which contains all possible combinations of \({x_3},{x_4}\) and \({x_5}\) and four rows \({x_1}{x_2},{x_1}{\bar x_2},{\bar x_1}{\bar x_2}\) and \({\bar x_1}{x_2}\); which contains all possible combinations of \({x_1}\) and \({x_2}\).

Need to place a \(1\) in all cells corresponding with \({x_2}{x_4}\), which are the cells in the row \({x_1}{x_2}/{\bar x_1}{x_2}\) and in the columns \({x_3}{x_4}{x_5}/{\bar x_3}{x_4}{x_5}/{x_3}{x_4}{\bar x_5}/{\bar x_3}{x_4}{\bar x_5}\)

05

Step 5:Placing the values in the cells

A \(K{\bf{ - }}\)map for a function in five variables is a table with \(8\) columns \({x_3}{x_4}{x_5},{x_3}{x_4}{\bar x_5},{x_3}{\bar x_4}{\bar x_5},{x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{\bar x_5},{\bar x_3}{x_4}{\bar x_5}\) and \({\bar x_3}{x_4}{x_5}\); which contains all possible combinations of \({x_3},{x_4}\) and \({x_5}\) and four rows \({x_1}{x_2},{x_1}{\bar x_2},{\bar x_1}{\bar x_2}\) and \({\bar x_1}{x_2}\); which contains all possible combinations of \({x_1}\) and \({x_2}\).

Need to place a \(1\) in all cells corresponding with \({\bar x_3}{\bar x_4}\), which are the cells in the row \({x_1}{x_2}/{\bar x_1}{x_2}/{x_1}{\bar x_2}/{\bar x_1}{\bar x_2}\) and in the columns \({\bar x_3}{\bar x_4}{x_5}/{\bar x_3}{\bar x_4}{\bar x_5}\)

06

Placing the values in the cells

A \(K{\bf{ - }}\)map for a function in five variables is a table with \(8\) columns \({x_3}{x_4}{x_5},{x_3}{x_4}{\bar x_5},{x_3}{\bar x_4}{\bar x_5},{x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{\bar x_5},{\bar x_3}{x_4}{\bar x_5}\) and \({\bar x_3}{x_4}{x_5}\); which contains all possible combinations of \({x_3},{x_4}\) and \({x_5}\) and four rows \({x_1}{x_2},{x_1}{\bar x_2},{\bar x_1}{\bar x_2}\) and \({\bar x_1}{x_2}\); which contains all possible combinations of \({x_1}\) and \({x_2}\).

Need to place a \(1\) in all cells corresponding with \({x_3}\), which are the cells in the row \({x_1}{x_2}/{\bar x_1}{x_2}/{x_1}{\bar x_2}/{\bar x_1}{\bar x_2}\) and in the columns \({x_3}{x_4}{x_5}/{x_3}{x_4}{\bar x_5}/{x_3}{\bar x_4}{x_5}/{x_3}{\bar x_4}{\bar x_5}\).

07

Placing the values in the cells

A \(K{\bf{ - }}\)map for a function in five variables is a table with \(8\) columns \({x_3}{x_4}{x_5},{x_3}{x_4}{\bar x_5},{x_3}{\bar x_4}{\bar x_5},{x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{x_5},{\bar x_3}{\bar x_4}{\bar x_5},{\bar x_3}{x_4}{\bar x_5}\) and \({\bar x_3}{x_4}{x_5}\); which contains all possible combinations of \({x_3},{x_4}\) and \({x_5}\) and four rows \({x_1}{x_2},{x_1}{\bar x_2},{\bar x_1}{\bar x_2}\) and \({\bar x_1}{x_2}\); which contains all possible combinations of \({x_1}\) and \({x_2}\).

Need to place a \(1\) in all cells corresponding with \({\bar x_5}\), which are the cells in the row \({x_1}{x_2}/{\bar x_1}{x_2}/{x_1}{\bar x_2}/{\bar x_1}{\bar x_2}\) and in the columns \({x_3}{x_4}{\bar x_5}/{\bar x_3}{x_4}{\bar x_5}/{x_3}{\bar x_4}{\bar x_5}/{\bar x_3}{\bar x_4}{\bar x_5}\).

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Most popular questions from this chapter

How many cells in a \({\bf{K}}\)-map for Boolean functions with six variables are needed to represent \({{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{\bar x}}_{\bf{1}}}{{\bf{x}}_{\bf{6}}}{\bf{, }}{{\bf{\bar x}}_{\bf{1}}}{{\bf{x}}_{\bf{2}}}{{\bf{\bar x}}_{\bf{6}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{{\bf{x}}_{\bf{3}}}{{\bf{x}}_{\bf{4}}}{{\bf{x}}_{\bf{5}}}\), and \({{\bf{x}}_{\bf{1}}}{{\bf{\bar x}}_{\bf{2}}}{{\bf{x}}_{\bf{4}}}{{\bf{\bar x}}_{\bf{5}}}\), respectively\({\bf{?}}\)

Show that the set of operators \(\left\{ {{\bf{ + , \cdot}}} \right\}\) is not functionally complete.

Find a Boolean sum containing either x or \(\overline {\bf{x}} \), either y or \(\overline {\bf{y}} \), and either z or \(\overline {\bf{z}} \) that has the value 0 if and only if

a) \({\bf{x = }}\,{\bf{y = 1,}}\,{\bf{z = 0}}\)

b) \({\bf{x = }}\,{\bf{y = }}\,{\bf{z = 0}}\)

c) \({\bf{x = }}\,{\bf{z = 0,}}\,{\bf{y = 1}}\)

For each of these equalities either prove it is an identity or find a set of values of the variables for which it does not hold.

\(\begin{array}{l}a)x|(y\mid z){\bf{ = }}(x\mid y)|z\\b)x \downarrow (y \downarrow z){\bf{ = }}(x \downarrow y) \downarrow (x \downarrow z)\\c)x \downarrow (y\mid z){\bf{ = }}(x \downarrow y)\mid (x \downarrow z)\end{array}\)

Define the Boolean operator \( \odot \) as follows: \(1 \odot 1{\bf{ = }}1,1 \odot 0{\bf{ = }}0,0 \odot 1{\bf{ = }}0\), and \(0 \odot 0{\bf{ = }}1\).

Exercises 14-23 deal with the Boolean algebra \(\left\{ {{\bf{0,1}}} \right\}\) with addition,multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example \(8\).

22. Verify the unit property.
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