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

Let us evaluate each option using Boolean algebra properties:

• Option A: $\overline{P \oplus Q} = P \odot Q$. This is the standard definition of the XNOR operation (the complement of XOR). This is correct.

• Option B: $\bar{P} \oplus Q = \bar{P}\bar{Q} + \overline{\bar{P}}Q = \bar{P}\bar{Q} + PQ = P \odot Q$. This is correct.

• Option C: $\bar{P} \oplus \bar{Q} = \bar{P}\overline{\bar{Q}} + \overline{\bar{P}}\bar{Q} = \bar{P}Q + P\bar{Q} = P \oplus Q$. This is correct.

• Option D:

  • Left-hand side (LHS): $(P \oplus \bar{P}) \oplus Q = 1 \oplus Q = \bar{Q}$

  • Right-hand side (RHS): $(P \odot \bar{P}) \odot \bar{Q} = 0 \odot \bar{Q} = \overline{0 \oplus \bar{Q}} = \overline{\bar{Q}} = Q$

  • Since $\bar{Q} \neq Q$ in general, LHS $\neq$ RHS.

Therefore, statement D is NOT CORRECT.