contestada

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.

Respuesta :

MrRoyal

Answer:

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

Explanation:

Given

[tex]A\ \&\&\ B[/tex]

[tex]!A\ \&\&\ !B[/tex]

Required

Select the true statement

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

(a): [tex]A = true[/tex] and [tex]B = true[/tex]

[tex]A\ \&\&\ B[/tex]

[tex]true\ \&\&\ true \to true[/tex] i.e. true and true is true

[tex]!A\ \&\&\ !B[/tex]

[tex]!true\ \&\&\ !true[/tex]

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[tex]!true = false[/tex]

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So, we have:

[tex]false\ \&\&\ false \to false[/tex]

So:

[tex]A =true[/tex]  and [tex]B = true[/tex]

[tex]A\ \&\&\ B = true[/tex]

[tex]!A\ \&\&\ !B = false[/tex]

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

(b): [tex]A = true[/tex] and [tex]B = false[/tex]

[tex]A\ \&\&\ B[/tex]

[tex]true\ \&\&\ false \to false[/tex]

[tex]!A\ \&\&\ !B[/tex]

[tex]!true\ \&\&\ !false[/tex]

Solve each negation

[tex]false\ \&\&\ true \to false[/tex]

So:

[tex]A =true[/tex]  and [tex]B = false[/tex]

[tex]A\ \&\&\ B = false[/tex]

[tex]!A\ \&\&\ !B = false[/tex]

Hence, option (c) is incorrect

(c): [tex]A = false[/tex] and [tex]B = true[/tex]

[tex]A\ \&\&\ B[/tex]

[tex]false\ \&\&\ true \to false[/tex]

[tex]!A\ \&\&\ !B[/tex]

[tex]!false\ \&\&\ !true[/tex]

[tex]truee\ \&\&\ false \to false[/tex]

So:

[tex]A =false[/tex]   and [tex]B = true[/tex]

[tex]A\ \&\&\ B = false[/tex]

[tex]!A\ \&\&\ !B = false[/tex]

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.

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