Question

For Boolean variables A and B, \( A \oplus B = A\bar{B} + \bar{A}B \). Then, which of the following statements is/are correct?

• A. \( 1 \oplus A = \bar{A} \)
• B. \( A \oplus A = 0 \)
• C. \( 0 \oplus A = 0 \)
• D. \( A \oplus \bar{A} = 1 \)

Hint:

XOR is a "difference detector". It outputs 1 only when the two inputs are different:
- \( A \oplus A \): Inputs are identical, so output is always 0 (Statement B is correct).
- \( A \oplus \bar{A} \): Inputs are always opposite, so output is always 1 (Statement D is correct).
- \( 1 \oplus A \): Output is the invert of \( A \) (Statement A is correct).
This conceptual shortcut avoids algebraic derivations.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The \(\oplus\) operator denotes the Exclusive OR (XOR) logic function.
The characteristic of XOR is that it gives a high output (1) only when the inputs are different, and a low output (0) when the inputs are the same.
We can verify the properties by substituting the values into the provided Boolean expression \(A\overline{B} + \overline{A}B\).
Step 2: Key Formula or Approach:
XOR Identity: \(X \oplus Y = X\overline{Y} + \overline{X}Y\).
Basic identities: \(A \cdot 1 = A, A \cdot 0 = 0, A + 0 = A\).
Step 3: Detailed Explanation:
Step A: Testing \(1 \oplus A\).
Using the definition: \(1 \cdot \overline{A} + \overline{1} \cdot A\).
Since \(\overline{1} = 0\):
\(1 \cdot \overline{A} + 0 \cdot A = \overline{A} + 0 = \overline{A}\).
(Option A is correct).
Step B: Testing \(A \oplus A\).
Using the definition: \(A \cdot \overline{A} + \overline{A} \cdot A\).
Since \(A \cdot \overline{A} = 0\):
\(0 + 0 = 0\).
(Option B is correct).
Step C: Testing \(0 \oplus A\).
Using the definition: \(0 \cdot \overline{A} + \overline{0} \cdot A\).
Since \(\overline{0} = 1\):
\(0 + 1 \cdot A = A\).
(Option C is wrong as it says 0).
Step D: Statement (D) is \(A \oplus \overline{A} = 1\)? Actually, the OCR/image for (D) looks like \(A \oplus A = 1\).
As proven in Step B, \(A \oplus A = 0\), so (D) is incorrect.
Step 4: Final Answer:
By applying basic Boolean algebraic properties to the XOR definition, the identities \(1 \oplus A = \overline{A}\) and \(A \oplus A = 0\) are confirmed as correct.