Chapter 3, Problem 36E

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 K-Map using Boolean identities:

 Given:

 Solution:

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

 The following steps are used to obtain the Boolean expressions using Boolean identities.

 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. Here “X” variable denotes don’t care which means the user will take the “X” value as either 0 or 1.

 The below expression is obtained from the K-Map

 $\begin{array}{c}\text{F = x'}\cdot \text{y'}\cdot \text{z' + x'}\cdot \text{y'}\cdot \text{z + x}\cdot \text{y'}\cdot \text{z' + x}\cdot \text{y'}\cdot \text{z}\\ \text{= x'}\cdot \text{y'}\cdot \left(\text{z' + z}\right)\text{+ x}\cdot \text{y'}\cdot \left(\text{z' + z}\right)\left(\text{by Distribution law}\right)\\ =\text{x'}\cdot \text{y'}\cdot \left(\text{1}\right)\text{+ x}\cdot \text{y'}\cdot \left(\text{1}\right)\left(\text{by Inverse law}\right)\\ =\text{x'}\cdot \text{y'+ x}\cdot \text{y' }\left(\text{by Identity law}\right)\end{array}$

    $\begin{array}{l}\text{= }\left(\text{x' + x}\right)\cdot \text{y' }\left(\text{by Distribution law}\right)\\ \text{= }\left(\text{1}\right)\cdot \text{y' }\left(\text{by Inverse law}\right)\\ \text{= y' }\left(\text{by Identity law}\right)\end{array}$

 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. Here “X” variable denotes don’t care which means the user will take the “X” value as either 0 or 1.

 The below expression is obtained from the K-Map

 $\begin{array}{c}\text{F = x'}\cdot \text{y'}\cdot \text{z' + x}\cdot \text{y'}\cdot \text{z' + x'}\cdot \text{y}\cdot \text{z' + x}\cdot \text{y}\cdot \text{z'}\\ \text{= y'}\cdot \text{z'}\cdot \left(\text{x' + x}\right)\text{+ y}\cdot \text{z'}\cdot \left(\text{x' + x}\right)\left(\text{by Distribution law}\right)\\ =\text{y'}\cdot \text{z'}\cdot \left(\text{1}\right)\text{+ y}\cdot \text{z'}\cdot \left(\text{1}\right)\left(\text{by Inverse law}\right)\\ =\text{y'}\cdot \text{z'+ y}\cdot \text{z' }\left(\text{by Identity law}\right)\end{array}$

    $\begin{array}{l}\text{= }\left(\text{y' + y}\right)\cdot \text{z' }\left(\text{by Distribution law}\right)\\ \text{= }\left(\text{1}\right)\cdot \text{z' }\left(\text{by Inverse law}\right)\\ \text{= z' }\left(\text{by Identity law}\right)\end{array}$

 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. Here “X” variable denotes don’t care which means the user will take the “X” value as either 0 or 1.

 The below expression is obtained from the K-Map

 $\begin{array}{c}\text{F = x}\cdot \text{y'}\cdot \text{z' + x}\cdot \text{y'}\cdot \text{z + x}\cdot \text{y}\cdot \text{z + x}\cdot \text{y}\cdot \text{z'}\\ \text{= x}\cdot \text{y'}\cdot \left(\text{z' + z}\right)\text{+ x}\cdot \text{y}\cdot \left(\text{z' + z}\right)\left(\text{by Distribution law}\right)\\ =\text{x}\cdot \text{y'}\cdot \left(\text{1}\right)\text{+ x}\cdot \text{y}\cdot \left(\text{1}\right)\left(\text{by Inverse law}\right)\\ =\text{x}\cdot \text{y'+ x}\cdot \text{y }\left(\text{by Identity law}\right)\end{array}$

    $\begin{array}{l}\text{= }\left(\text{y' + y}\right)\cdot \text{x }\left(\text{by Distribution law}\right)\\ \text{= }\left(\text{1}\right)\cdot \text{x }\left(\text{by Inverse law}\right)\\ \text{= x }\left(\text{by Identity law}\right)\end{array}$

 Step4:

 Group all the expressions

 $\text{F}=\text{y' + z' + x}$

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

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 K-Map using Boolean identities:

 Given:

 Solution:

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

 The following steps are used to obtain the Boolean expressions using Boolean identities.

 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. Here “X” variable denotes don’t care which means the user will take the “X” value as either 0 or 1.

 The below expression is obtained from the K-Map

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

    $\begin{array}{l}\text{= w'}\cdot \text{x'}\cdot \left(\text{y' + y}\right)\left(\text{by Distribution law}\right)\\ \text{= w'}\cdot \text{x'}\cdot \left(\text{1}\right)\left(\text{by Inverse law}\right)\\ \text{= w'}\cdot \text{x' }\left(\text{by Identity law}\right)\end{array}$

 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. Here “X” variable denotes don’t care which means the user will take the “X” value as either 0 or 1.

 The below expression is obtained from the K-Map

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

    $\begin{array}{l}\text{= x'}\cdot \text{z'}\cdot \left(\text{w' + w}\right)\left(\text{by Distribution law}\right)\\ \text{= x'}\cdot \text{z'}\cdot \left(\text{1}\right)\left(\text{by Inverse law}\right)\\ \text{= x'}\cdot \text{z' }\left(\text{by Identity law}\right)\end{array}$

 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. Here “X” variable denotes don’t care which means the user will take the “X” value as either 0 or 1.

 The below expression is obtained from the K-Map

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

    $\begin{array}{l}\text{= w'}\cdot \text{z}\cdot \left(\text{x' + x}\right)\left(\text{by Distribution law}\right)\\ \text{= w'}\cdot \text{z}\cdot \left(\text{1}\right)\left(\text{by Inverse law}\right)\\ \text{= w'}\cdot \text{z }\left(\text{by Identity law}\right)\end{array}$

 Step4:

 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. Here “X” variable denotes don’t care which means the user will take the “X” value as either 0 or 1.

 The below expression is obtained from the K-Map

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

    $\begin{array}{l}\text{= w'}\cdot \text{y}\cdot \left(\text{x' + x}\right)\left(\text{by Distribution law}\right)\\ \text{= w'}\cdot \text{y}\cdot \left(\text{1}\right)\left(\text{by Inverse law}\right)\\ \text{= w'}\cdot \text{y }\left(\text{by Identity law}\right)\end{array}$

 Step5:

 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. Here “X” variable denotes don’t care which means the user will take the “X” value as either 0 or 1.

 The below expression is obtained from the K-Map

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

    $\begin{array}{l}\text{= y}\cdot \text{z}\cdot \left(\text{w' + w}\right)\left(\text{by Distribution law}\right)\\ \text{= y}\cdot \text{z}\cdot \left(\text{1}\right)\left(\text{by Inverse law}\right)\\ \text{= y}\cdot \text{z }\left(\text{by Identity law}\right)\end{array}$

 Step6:

 Group all the expressions

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

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