Question

Consider the following Boolean expressions.

A && B
!A && !B
Which of the following best describes the relationship between values produced by expression I and expression II?
(A) Expression I and expression II evaluate to different values for all values of A and B.
(B) Expression and expression II evaluate to the same value for all values of A and B.
(C) Expression and expression II evaluate to the same value only when A and B are the same.
(D) Expression and expression Il evaluate to the same value only when A and B differ.
(E) Expression I and expression Il evaluate to the same value whenever A is true.

Answer 1

(D) Expression I and expression Il evaluate to the same value only when A and B differ.

Step-by-step explanation:

Given

`A\ \&\&\ B`

`!A\ \&\&\ !B`

Required

Select the true statement

To do this, I will create the following case scenarios.

(a):
`A = true` and
`B = true`

`A\ \&\&\ B`

`true\ \&\&\ true \to true` i.e. true and true is true

`!A\ \&\&\ !B`

`!true\ \&\&\ !true`

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`!true = false`

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So, we have:

`false\ \&\&\ false \to false`

So:

`A =true` and
`B = true`

`A\ \&\&\ B = true`

`!A\ \&\&\ !B = false`

Hence, options (B) and (E) are incorrect

(b):
`A = true` and
`B = false`

`A\ \&\&\ B`

`true\ \&\&\ false \to false`

`!A\ \&\&\ !B`

`!true\ \&\&\ !false`

Solve each negation

`false\ \&\&\ true \to false`

So:

`A =true` and
`B = false`

`A\ \&\&\ B = false`

`!A\ \&\&\ !B = false`

Hence, option (c) is incorrect

(c):
`A = false` and
`B = true`

`A\ \&\&\ B`

`false\ \&\&\ true \to false`

`!A\ \&\&\ !B`

`!false\ \&\&\ !true`

`truee\ \&\&\ false \to false`

So:

`A =false` and
`B = true`

`A\ \&\&\ B = false`

`!A\ \&\&\ !B = false`

Case scenarios b and c implies that option (d) is correct because different values of A and B gives the same value of both expression which is false

This also implies that (a) is incorrect.