Question

If \( x \) and \( y \) are Boolean variables, then which of the following statement is/are incorrect?
a. \( x \lor 0 = x, \, x \land 1 = x \)
b. \( x \lor x' = 1, \, x \land x' = 0 \)
c. \( x \land (x \lor y) = x \)
d. \( x \land (x \land y) = x \lor y \)

• A. a, b and c
• B. a only
• C. c only
• D. d only

Hint:

For Boolean algebra, remember the absorption and complement laws: - Absorption Law: \( x \land (x \lor y) = x \) - Complement Law: \( x \lor x' = 1 \), \( x \land x' = 0 \)

Answer 1

The Correct Option is C

Step 1: Analyzing statement (a)
For Boolean operations: \[ x \lor 0 = x \quad \text{and} \quad x \land 1 = x \] These statements are correct, as:
The OR operation with 0 leaves the variable unchanged.
The AND operation with 1 leaves the variable unchanged.
Step 2: Analyzing statement (b) For Boolean operations: \[ x \lor x' = 1 \quad \text{and} \quad x \land x' = 0 \] These are correct because:
The OR operation between a variable and its complement results in 1.
The AND operation between a variable and its complement results in 0.
Step 3: Analyzing statement (c) For Boolean operations: \[ x \land (x \lor y) = x \] This statement is true by the absorption law, which states that \( x \land (x \lor y) = x \). So, this statement is correct. Step 4: Analyzing statement (d) For Boolean operations: \[ x \land (x \land y) = x \lor y \] This statement is incorrect. The correct simplification would be: \[ x \land (x \land y) = x \land y \] This does not simplify to \( x \lor y \). Therefore, statement (d) is incorrect. Thus, the only incorrect statement is (c), making the correct answer (3) c only.