Chapter 3, Problem 34E

Program Plan Intro

a.

K-Map:

• K-Map stands for Karnaugh Map which is used to reduce the logic functions more easily and quickly.
• It will minimize the Boolean expressions without using Boolean algebra theorems.
• 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:

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

 The K-Map for the given expression is as follows:

 The following steps are used to obtain the simplified 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.

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

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

 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.

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

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

 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.

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

 Step4:

 Group all the expressions

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

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

Program Plan Intro

b.

K-Map:

• K-Map stands for Karnaugh Map which is used to reduce the logic functions more easily and quickly.
• It will minimize the Boolean expressions without using Boolean algebra theorems.
• 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:

 $\text{F}\left(\text{x, y, z}\right)=\text{w'}\cdot \text{x'}\cdot \text{y'}\cdot \text{z' + w'}\cdot \text{z + w'}\cdot \text{x'}\cdot \text{y}\cdot \text{z' + w'}\cdot \text{x}\cdot \text{y'}\cdot \text{z' + w}\cdot \text{x'}\cdot \text{y}$

 Simplifying the given equation into 4 variable terms as shown below:

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

                 $\begin{array}{c}\text{ w'}\cdot \text{x}\cdot \text{y'}\cdot \text{z' + w}\cdot \text{x'}\cdot \text{y}\cdot \text{z' + w}\cdot \text{x'}\cdot \text{y}\cdot \text{z}\\ \text{= w'}\cdot \text{x'}\cdot \text{y'}\cdot \text{z' + w'}\cdot \text{x'}\cdot \text{y'}\cdot \text{z + w'}\cdot \text{x'}\cdot \text{y}\cdot \text{z +}\\ \text{ w'}\cdot \text{x}\cdot \text{y'}\cdot \text{z + w'}\cdot \text{x}\cdot \text{y}\cdot \text{z + w'}\cdot \text{x'}\cdot \text{y}\cdot \text{z' +}\\ \text{ w'}\cdot \text{x}\cdot \text{y'}\cdot \text{z' + w}\cdot \text{x'}\cdot \text{y}\cdot \text{z' + w}\cdot \text{x'}\cdot \text{y}\cdot \text{z}\end{array}$

 The K-Map for the given expression is as follows:

 The following steps are used to obtain the simplified 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.

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

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

 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.

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

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

 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.

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

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

 Step4:

 Group all the expressions

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

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

Program Plan Intro

c.

K-Map:

• K-Map stands for Karnaugh Map which is used to reduce the logic functions more easily and quickly.
• It will minimize the Boolean expressions without using Boolean algebra theorems.
• 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:

 $\text{F}\left(\text{x, y, z}\right)=\text{w'}\cdot \text{x'}\cdot \text{y' + w'}\cdot \text{x}\cdot \text{z + w}\cdot \text{x}\cdot \text{z + w}\cdot \text{x'}\cdot \text{y'}\cdot \text{z}\text{'}$

 Simplifying the given equation into 4 variable terms as shown below:

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

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

 The K-Map for the given expression is as follows:

 The following steps are used to obtain the simplified 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.

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

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

 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.

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

 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.

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

 Step4:

 Group all the expressions

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

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