Question

Given the Boolean expression \( (A \oplus B) \land (B \rightarrow C) \), which of the following rows in the truth table would have an output of 1 (True)?

• A. \( A = 1, B = 1, C = 1 \)
• B. \( A = 0, B = 1, C = 0 \)
• C. \( A = 0, B = 0, C = 0 \)
• D. \( A = 1, B = 0, C = 1 \)

Hint:

When dealing with Boolean expressions involving XOR and implication, break them down into simpler operations and evaluate step-by-step.

Answer 1

The Correct Option is D

We are given the Boolean expression: \[ (A \oplus B) \land (B \rightarrow C) \] Let's break this down: 1. XOR operation (\( A \oplus B \)): This will be true if \( A \) and \( B \) have opposite values. \[ A \oplus B = 1 \text{if} A \neq B, \text{else} 0 \] 2. Implication (\( B \rightarrow C \)): This will be true unless \( B = 1 \) and \( C = 0 \). \[ B \rightarrow C = 1 \text{if} B = 0 \text{or} C = 1, \text{else} 0 \] Now let's evaluate the truth table for each option: 

- Option (a): \( A = 1, B = 1, C = 1 \) - \( A \oplus B = 0 \) (since \( A = B \)) - \( B \rightarrow C = 1 \) (since \( C = 1 \)) 

- The result is \( 0 \land 1 = 0 \) 

- Option (b): \( A = 0, B = 1, C = 0 \) - \( A \oplus B = 1 \) (since \( A \neq B \)) - \( B \rightarrow C = 0 \) (since \( B = 1 \) and \( C = 0 \)) 

- The result is \( 1 \land 0 = 0 \) 

- Option (c): \( A = 0, B = 0, C = 0 \) - \( A \oplus B = 0 \) (since \( A = B \)) - \( B \rightarrow C = 1 \) (since \( B = 0 \)) 

- The result is \( 0 \land 1 = 0 \) 

- Option (d): \( A = 1, B = 0, C = 1 \) - \( A \oplus B = 1 \) (since \( A \neq B \)) - \( B \rightarrow C = 1 \) (since \( B = 0 \)) 

- The result is \( 1 \land 1 = 1 \) 

Thus, the correct answer is \( \boxed{(d) \, A = 1, B = 0, C = 1} \).