Chapter 3, Problem 24E

Program Description Answer

Therefore, the expression in (b) is not logically equivalent.

Hence, the correct option is “B”.

Explanation of Solution

Explanation for correct option:

The given problem is solved by two methods, first method is by simplifying the given expressions and finally finds out the odd one and the second method is by constructing the truth table. The second method is not efficient to solve this problem so let solve this problem using first method.

b. Given expression:

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

The given expression is can’t be simplified because, all the four variables are unique.

Explanation for incorrect option:

a. Given expression:

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

Taking “w” common from all the terms

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

c. Given expression:

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

Keep this expression as it is for comparison purpose

d. Given expression:

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

Simplification:

$\begin{array}{l}\text{w}\cdot \text{x'}\cdot \text{y}\cdot \text{z' + w}\cdot \text{x'}\cdot \text{y' + w}\cdot \text{y'}\cdot \text{z' + w}\cdot \text{z}\\ \text{= w}\cdot \text{x'}\cdot \left(\text{y}\cdot \text{z' + y'}\right)\text{ + w}\cdot \left(\text{y'}\cdot \text{z' + z}\right)\left(\text{by Distribution law}\right)\\ =\text{ w}\cdot \text{x'}\cdot \left(\text{y' + z'}\right)\text{ + w}\cdot \left(\text{z + y'}\right)\left(\text{by Redundant Literal rule}\right)\\ =\text{ w}\cdot \text{x'}\cdot \text{y' + w}\cdot \text{x'}\cdot \text{z' + w}\cdot \text{z + w}\cdot \text{y' }\left(\text{by Distribution law}\right)\end{array}$

$\begin{array}{l}=\text{ w}\cdot \text{y' }\left(\text{x' + 1}\right)\text{+ w}\cdot \text{x'}\cdot \text{z' + w}\cdot \text{z }\left(\text{by Distribution law}\right)\\ =\text{ w}\cdot \text{y'}\left(\text{1}\right)\text{+ w}\cdot \text{x'}\cdot \text{z' + w}\cdot \text{z }\left(\text{by Null law}\right)\\ =\text{ w}\cdot \text{y' + w}\cdot \left(\text{x'}\cdot \text{z' + z}\right)\left(\text{by Distribution law}\right)\\ =\text{ w}\cdot \text{y' + w}\cdot \left(\text{z + x'}\right)\left(\text{by Redundant Literal rule}\right)\end{array}$

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

From the simplification of the expressions (a), (c) and (d) are logically equivalent because all the three expressions give same result.

Therefore, the expressions (a), (c) and (d) are not a correct option.