GATE CS 2018 Q.14 (CS Q.4): Boolean Algebra and Logic Minimization

Let $\oplus$ and $\odot$ denote the Exclusive OR and Exclusive NOR operations, respectively. Which one of the following is NOT CORRECT?

$\overline{P \oplus Q} = P \odot Q$

$\overline{P} \oplus Q = P \odot Q$

$\overline{P} \oplus \overline{Q} = P \oplus Q$

$(P \oplus \overline{P}) \oplus Q = (P \odot \overline{P}) \odot \overline{Q}$

Official answer and solution

Solution

Evaluate each Boolean expression algebraically using definitions of XOR and XNOR.

Recall the definitions:

A \oplus B = A\overline{B} + \overline{A}B

A \odot B = AB + \overline{A}\,\overline{B} = \overline{A \oplus B}

Check each statement:

(A) $\overline{P \oplus Q} = P \odot Q$ : True by definition of XNOR.
(B) $\overline{P} \oplus Q = \overline{P}\,\overline{Q} + P Q = P \odot Q$ : True.
(C) $\overline{P} \oplus \overline{Q} = \overline{P} Q + P\overline{Q} = P \oplus Q$ : True.
(D) LHS: $(P \oplus \overline{P}) \oplus Q = 1 \oplus Q = \overline{Q}$ .
RHS: $(P \odot \overline{P}) \odot \overline{Q} = 0 \odot \overline{Q}$ . Since $0 \odot X = 0 \cdot X + 1 \cdot \overline{X} = \overline{X}$ , with $X = \overline{Q}$ we get $0 \odot \overline{Q} = \overline{\overline{Q}} = Q$ .
Because $\overline{Q} \neq Q$ , equation (D) is NOT correct.

Final answer

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