Let $\oplus$ denote the Exclusive OR (XOR) operation. Let ‘1’ and ‘0’ denote the binary constants. Consider the following Boolean expression for $F$ over two variables $P$ and $Q$: $$F(P, Q) = ((1 \oplus P) \oplus (P \oplus Q)) \oplus ((P \oplus Q) \oplus (Q \oplus 0))$$ The equivalent expression for $F$ is

We use the standard algebraic properties of the Exclusive OR ($\oplus$) operator:

1. Associativity and Commutativity: $A \oplus B = B \oplus A$ and $(A \oplus B) \oplus C = A \oplus (B \oplus C)$.

2. Self-inverse: $A \oplus A = 0$.

3. Identity: $A \oplus 0 = A$.

4. Inversion: $A \oplus 1 = \overline{A}$.

Let us simplify the two major parts of the expression:

• First term:

  $$T_1 = (1 \oplus P) \oplus (P \oplus Q) = 1 \oplus (P \oplus P) \oplus Q = 1 \oplus 0 \oplus Q = 1 \oplus Q = \overline{Q}$$

• Second term:

  $$T_2 = (P \oplus Q) \oplus (Q \oplus 0) = P \oplus (Q \oplus Q) \oplus 0 = P \oplus 0 \oplus 0 = P$$

Now combine the two terms:

Hence, the equivalent expression is $\overline{P \oplus Q}$, which corresponds to option D.