Consider the following Boolean expression of a function $F$ : $$F(P, Q) = (\bar{P} + Q) \oplus (\bar{P}Q)$$ Which of the following expressions is/are equivalent to $F$ ?

Given the function:

Let $A = \bar{P} + Q$ and $B = \bar{P}Q$. Recall the definition of XOR: $A \oplus B = A\bar{B} + \bar{A}B$.

1. Calculate $A\bar{B}$:

   $$\bar{B} = \overline{\bar{P}Q} = P + \bar{Q}$$

   $$A\bar{B} = (\bar{P} + Q)(P + \bar{Q}) = \bar{P}P + \bar{P}\bar{Q} + QP + Q\bar{Q} = 0 + \bar{P}\bar{Q} + PQ + 0 = \bar{P}\bar{Q} + PQ$$

2. Calculate $\bar{A}B$:

   $$\bar{A} = \overline{\bar{P} + Q} = P\bar{Q}$$

   $$\bar{A}B = (P\bar{Q})(\bar{P}Q) = P\bar{P}\bar{Q}Q = 0$$

3. Combine terms:

   $$F(P, Q) = (\bar{P}\bar{Q} + PQ) + 0 = \bar{P}\bar{Q} + PQ$$

   This is the standard XNOR function, which can be written as:

   • $F(P, Q) = \overline{P \oplus Q}$ (Option A)

   • $F(P, Q) = \bar{P} \oplus Q$ (Option C)

Options B and D represent XOR ($P \oplus Q$ and $\bar{P} \oplus \bar{Q} = P \oplus Q$), which are the negation of $F$.

Hence, the correct options are A and C.