Q151 / 155 · WBJEE 2026 · Physics
For Boolean variables $A$ and $B$, $A \oplus B = A\bar{B} + \bar{A}B$. Then, which of the following statements is/are correct?
- Correct answerA$1 \oplus A = \bar{A}$
- Correct answerB$A \oplus A = 0$
- C$0 \oplus A = 0$
- Correct answerD$A \oplus \bar{A} = 1$
Solution
Given $A \oplus B = A\bar{B} + \bar{A}B$
Evaluating the first expression:
$1 \oplus A = 1 \cdot \bar{A} + \bar{1} \cdot A = \bar{A} + 0 = \bar{A}$
Evaluating the second expression:
$A \oplus A = A \cdot \bar{A} + \bar{A} \cdot A = 0 + 0 = 0$
Evaluating the third expression:
$0 \oplus A = 0 \cdot \bar{A} + \bar{0} \cdot A = 0 + 1 \cdot A = A$
Evaluating the fourth expression:
$A \oplus \bar{A} = A \cdot \overline{\bar{A}} + \bar{A} \cdot \bar{A} = A \cdot A + \bar{A} = A + \bar{A} = 1$
The first, second, and fourth expressions are correct.
Answer: $1 \oplus A = \bar{A}$; $A \oplus A = 0$; $A \oplus \bar{A} = 1$
Evaluating the first expression:
$1 \oplus A = 1 \cdot \bar{A} + \bar{1} \cdot A = \bar{A} + 0 = \bar{A}$
Evaluating the second expression:
$A \oplus A = A \cdot \bar{A} + \bar{A} \cdot A = 0 + 0 = 0$
Evaluating the third expression:
$0 \oplus A = 0 \cdot \bar{A} + \bar{0} \cdot A = 0 + 1 \cdot A = A$
Evaluating the fourth expression:
$A \oplus \bar{A} = A \cdot \overline{\bar{A}} + \bar{A} \cdot \bar{A} = A \cdot A + \bar{A} = A + \bar{A} = 1$
The first, second, and fourth expressions are correct.
Answer: $1 \oplus A = \bar{A}$; $A \oplus A = 0$; $A \oplus \bar{A} = 1$
Hence, the correct answer is options: 'A', 'B', 'D'.