Chapter 3, Problem 31E

Program Plan Intro

a.

K-Map:

• K-Map stands for Karnaugh Map which is used to reduce the logic functions more easily and quickly.
• By using K-Map, the Boolean expressions with two to four variables are easily reduced.

Explanation of Solution

Simplification of Boolean expression using K-Map:

 Given:

 Solution:

 The simplified expression using K-Map is $\text{F}=\text{w}\cdot \text{y}\cdot \text{z + x'}\cdot \text{y'}\cdot \text{z' + w'}\cdot \text{z'}$

 The following steps are used to obtain the Boolean expressions.

 Step1:

 The group the 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

     $\begin{array}{c}\text{F = w'}\cdot \text{x'}\cdot \text{y'}\cdot \text{z' + w'}\cdot \text{x}\cdot \text{y'}\cdot \text{z' + w'}\cdot \text{x'}\cdot \text{y}\cdot \text{z' + w'}\cdot \text{x}\cdot \text{y}\cdot \text{z'}\\ \text{= w'}\cdot \text{y'}\cdot \text{z'}\cdot \left(\text{x' + x}\right)\text{+ w'}\cdot \text{y}\cdot \text{z' }\cdot \left(\text{x' + x}\right)\\ =\text{w'}\cdot \text{y'}\cdot \text{z'}\cdot \left(\text{1}\right)\text{+ w'}\cdot \text{y}\cdot \text{z' }\cdot \left(\text{1}\right)\\ =\text{w'}\cdot \text{z' }\left(\text{y' + y}\right)\end{array}$

        $\begin{array}{c}=\text{w'}\cdot \text{z'}\left(1\right)\\ =\text{w'}\cdot \text{z'}\end{array}$

 Step2:

 Then group another possible 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

     $\begin{array}{c}\text{F = w}\cdot \text{x}\cdot \text{y}\cdot \text{z + w}\cdot \text{x'}\cdot \text{y}\cdot \text{z}\\ \text{= w}\cdot \text{y}\cdot \text{z}\cdot \left(\text{x + x'}\right)\\ =\text{w}\cdot \text{y}\cdot \text{z}\end{array}$

 Step3:

 Then group another possible 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

  $\begin{array}{c}\text{F = w'}\cdot \text{x'}\cdot \text{y'}\cdot \text{z' + w}\cdot \text{x'}\cdot \text{y'}\cdot \text{z'}\\ \text{= }\left(\text{w' + w}\right)\cdot \text{x'}\cdot \text{y'}\cdot \text{z'}\\ =\left(\text{1}\right)\cdot \text{x'}\cdot \text{y'}\cdot \text{z'}\\ =\text{x'}\cdot \text{y'}\cdot \text{z'}\end{array}$

 Step4:

 Group all the expressions

 $\text{F}=\text{w}\cdot \text{y}\cdot \text{z + x'}\cdot \text{y'}\cdot \text{z' + w'}\cdot \text{z'}$

 Therefore, the simplified expression is “$F=w\cdot y\cdot z+x\text{'}\cdot y\text{'}\cdot z\text{'}+w\text{'}\cdot z\text{'}$”.

Program Plan Intro

b.

K-Map:

• K-Map stands for Karnaugh Map which is used to reduce the logic functions more easily and quickly.
• By using K-Map, the Boolean expressions with two to four variables are easily reduced.

Explanation of Solution

Simplification of Boolean expression using K-Map:

 Given:

 Solution:

 The simplified expression using K-Map is $\text{F = w'}\cdot \text{x' + w'}\cdot \text{y + w}\cdot \text{x + w}\cdot \text{x'}\cdot \text{z'}$

 The following steps are used to obtain the Boolean expressions.

 Step1:

 Then group another possible 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

  $\begin{array}{c}\text{F = w}\cdot \text{x'}\cdot \text{y'}\cdot \text{z' + w}\cdot \text{x'}\cdot \text{y}\cdot \text{z'}\\ \text{= w}\cdot \text{x'}\cdot \text{z'}\cdot \left(\text{y + y'}\right)\\ =\text{w}\cdot \text{x'}\cdot \text{z'}\cdot \left(1\right)\\ =\text{w}\cdot \text{x'}\cdot \text{z'}\end{array}$

 Step2:

 Then group another possible 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

     $\begin{array}{c}\text{F = w}\cdot \text{x}\cdot \text{y'}\cdot \text{z' + w}\cdot \text{x}\cdot \text{y'}\cdot \text{z + w}\cdot \text{x}\cdot \text{y}\cdot \text{z + w}\cdot \text{x}\cdot \text{y}\cdot \text{z}\text{'}\\ \text{= w}\cdot \text{x}\cdot \text{y'}\cdot \left(\text{z + z'}\right)\text{ + w}\cdot \text{x}\cdot \text{y}\cdot \left(\text{z + z}\text{'}\right)\\ =\text{w}\cdot \text{x}\cdot \text{y' + w}\cdot \text{x}\cdot \text{y}\\ =\text{w}\cdot \text{x}\cdot \left(\text{y' + y}\right)\end{array}$

        $\begin{array}{c}=\text{w}\cdot \text{x}\cdot \left(\text{1}\right)\\ =\text{w}\cdot \text{x}\end{array}$

 Step3:

 The group the 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

  $\begin{array}{c}\text{F = w'}\cdot \text{x'}\cdot \text{y'}\cdot \text{z' + w'}\cdot \text{x'}\cdot \text{y'}\cdot \text{z + w'}\cdot \text{x'}\cdot \text{y}\cdot \text{z + w'}\cdot \text{x'}\cdot \text{y}\cdot \text{z}\text{'}\\ \text{= w'}\cdot \text{x'}\cdot \text{y'}\cdot \left(\text{z' + z}\right)\text{ + w'}\cdot \text{x'}\cdot \text{y}\cdot \left(\text{z + z}\text{'}\right)\\ =\text{w'}\cdot \text{x'}\cdot \text{y'}\cdot \left(\text{1}\right)\text{ + w'}\cdot \text{x'}\cdot \text{y}\cdot \left(1\right)\\ =\text{w'}\cdot \text{x'}\cdot \left(\text{y' + y}\right)\end{array}$

        $\begin{array}{c}=\text{w'}\cdot \text{x'}\cdot \left(\text{1}\right)\\ =\text{w'}\cdot \text{x'}\end{array}$

 Step4:

 Then group another possible 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

     $\begin{array}{c}\text{F = w'}\cdot \text{x'}\cdot \text{y}\cdot \text{z + w'}\cdot \text{x'}\cdot \text{y}\cdot \text{z' + w'}\cdot \text{x}\cdot \text{y}\cdot \text{z + w'}\cdot \text{x}\cdot \text{y}\cdot \text{z}\text{'}\\ \text{= w'}\cdot \text{x'}\cdot \text{y}\cdot \left(\text{z + z'}\right)\text{ + w'}\cdot \text{x}\cdot \text{y}\cdot \left(\text{z + z}\text{'}\right)\\ =\text{w'}\cdot \text{x'}\cdot \text{y + w'}\cdot \text{x}\cdot \text{y}\\ =\text{w'}\cdot \text{y}\cdot \left(\text{x' + x}\right)\end{array}$

            $\begin{array}{c}=\text{w'}\cdot \text{y}\cdot \left(\text{1}\right)\\ =\text{w'}\cdot \text{y}\end{array}$

 Step5:

 Group all the expressions

 $\text{F = w'}\cdot \text{x' + w'}\cdot \text{y + w}\cdot \text{x + w}\cdot \text{x'}\cdot \text{z' }$

 Therefore, the simplified expression is “$F=w\text{'}\cdot x\text{'}+w\text{'}\cdot y+w\cdot x+w\cdot x\text{'}\cdot z\text{'}$”.

Program Plan Intro

c.

K-Map:

• K-Map stands for Karnaugh Map which is used to reduce the logic functions more easily and quickly.
• By using K-Map, the Boolean expressions with two to four variables are easily reduced.

Explanation of Solution

Simplification of Boolean expression using K-Map:

 Given:

 Solution:

 The simplified expression using K-Map is $\text{F = y'}\cdot \text{z + w}\cdot \text{y' + w'}\cdot \text{x}\cdot \text{y + w'}\cdot \text{y}\cdot \text{z' + x'}\cdot \text{y}\cdot \text{z'}$

 The following steps are used to obtain the Boolean expressions.

 Step1:

 Then group another possible 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

     $\begin{array}{c}\text{F = w}\cdot \text{x}\cdot \text{y'}\cdot \text{z' + w}\cdot \text{x}\cdot \text{y'}\cdot \text{z + w}\cdot \text{x'}\cdot \text{y'}\cdot \text{z' + w}\cdot \text{x'}\cdot \text{y'}\cdot \text{z}\\ \text{= w}\cdot \text{x}\cdot \text{y'}\cdot \left(\text{z' + z}\right)\text{ + w}\cdot \text{x'}\cdot \text{y'}\cdot \left(\text{z + z'}\right)\\ =\text{w}\cdot \text{x}\cdot \text{y'}\cdot \left(\text{1}\right)\text{ + w}\cdot \text{x'}\cdot \text{y'}\cdot \left(\text{1}\right)\\ =\text{w}\cdot \text{x}\cdot \text{y' + w}\cdot \text{x'}\cdot \text{y'}\end{array}$

          $\begin{array}{c}=\text{w}\cdot \text{y'}\cdot \left(\text{x + x'}\right)\\ =\text{w}\cdot \text{y'}\cdot \left(\text{1}\right)\\ =\text{w}\cdot \text{y'}\end{array}$

 Step2:

 The group the 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

  $\begin{array}{c}\text{F = w'}\cdot \text{x'}\cdot \text{y'}\cdot \text{z + w'}\cdot \text{x}\cdot \text{y'}\cdot \text{z + w}\cdot \text{x}\cdot \text{y'}\cdot \text{z + w}\cdot \text{x'}\cdot \text{y'}\cdot \text{z}\\ \text{= w'}\cdot \text{y'}\cdot \text{z}\cdot \left(\text{x' + x}\right)\text{ + w}\cdot \text{y'}\cdot \text{z }\cdot \left(\text{x + x'}\right)\\ =\text{w'}\cdot \text{y'}\cdot \text{z}\cdot \left(\text{1}\right)\text{ + w}\cdot \text{y'}\cdot \text{z}\cdot \left(1\right)\\ =\text{w'}\cdot \text{y'}\cdot \text{z + w}\cdot \text{y'}\cdot \text{z}\end{array}$

           $\begin{array}{c}=\left(\text{w' + w}\right)\cdot \text{y'}\cdot \text{z}\\ =\left(\text{1}\right)\cdot \text{y'}\cdot \text{z}\\ =\text{y'}\cdot \text{z}\end{array}$

 Step3:

 Then group another possible 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

  $\begin{array}{c}\text{F = w'}\cdot \text{x}\cdot \text{y}\cdot \text{z + w'}\cdot \text{x}\cdot \text{y}\cdot \text{z'}\\ \text{= w'}\cdot \text{x}\cdot \text{y}\cdot \left(\text{z' + z}\right)\\ =\text{w'}\cdot \text{x}\cdot \text{y}\cdot \left(\text{1}\right)\\ =\text{w'}\cdot \text{x}\cdot \text{y}\end{array}$

 Step4:

 Then group another possible 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

     $\begin{array}{c}\text{F = w'}\cdot \text{x}\cdot \text{y}\cdot \text{z' + w'}\cdot \text{x'}\cdot \text{y}\cdot \text{z'}\\ \text{= w'}\cdot \text{y}\cdot \text{z'}\cdot \left(\text{x' + x}\right)\\ =\text{w'}\cdot \text{y}\cdot \text{z'}\cdot \left(\text{1}\right)\\ =\text{w'}\cdot \text{y}\cdot \text{z'}\end{array}$

 Step5:

 Then group another possible 1’s in the table. Then write the Boolean expression according to the mapping. Write 0 term’s as complemented variable like x’ and 1 term’s as it is like x.

     $\begin{array}{c}\text{F = w'}\cdot \text{x'}\cdot \text{y}\cdot \text{z' + w}\cdot \text{x'}\cdot \text{y}\cdot \text{z'}\\ \text{= x'}\cdot \text{y}\cdot \text{z'}\cdot \left(\text{w' + w}\right)\\ =\text{x'}\cdot \text{y}\cdot \text{z'}\cdot \left(\text{1}\right)\\ =\text{x'}\cdot \text{y}\cdot \text{z'}\end{array}$

 Step6:

 Group all the expressions

 $\text{F = y'}\cdot \text{z + w}\cdot \text{y' + w'}\cdot \text{x}\cdot \text{y + w'}\cdot \text{y}\cdot \text{z' + x'}\cdot \text{y}\cdot \text{z'}$

 Therefore, the simplified expression is “$F=y\text{'}\cdot z+w\cdot y\text{'}+w\text{'}\cdot x\cdot y+w\text{'}\cdot y\cdot z\text{'}+x\text{'}\cdot y\cdot z\text{'}$”.