Question 21

Evaluating $F(P, Q) = (\overline{P} + Q) \oplus (\overline{P}Q)$ for all four input combinations $(P, Q)$:

• $(0, 0)$: $(1 + 0) \oplus 0 = 1 \oplus 0 = 1$
• $(0, 1)$: $(1 + 1) \oplus 1 = 1 \oplus 1 = 0$
• $(1, 0)$: $(0 + 0) \oplus 0 = 0 \oplus 0 = 0$
• $(1, 1)$: $(0 + 1) \oplus 0 = 1 \oplus 0 = 1$

Thus $F(P, Q) = \overline{P}\,\overline{Q} + PQ = P \odot Q$, which is the XNOR function:

• Option (A): $\overline{P \oplus Q} = P \odot Q = F(P, Q)$.
• Option (C): $\overline{P} \oplus Q = (\overline{P})\overline{Q} + P Q = \overline{P}\,\overline{Q} + PQ = F(P, Q)$.
• Options (B) and (D) both equal $P \oplus Q \neq F(P, Q)$.