Discussion
Let \(\oplus\) and \(\odot\) denote the Exclusive OR and Exclusive NOR operations, respectively. Which one of the following is NOT CORRECT?
Answer: D
The incorrect identity is \((P \oplus \bar{P}) \oplus Q = (P \odot \bar{P}) \odot \bar{Q}\).
A variable and its complement always differ, so \(P \oplus \bar{P} = 1\) and \(P \odot \bar{P} = 0\). The left side is \(1 \oplus Q = \bar{Q}\). The right side is \(0 \odot \bar{Q}\), and XNOR with 0 complements its input, giving \(Q\). Since \(\bar{Q} \ne Q\), the identity fails.
The other three hold. XNOR is by definition the complement of XOR, so \(\overline{P \oplus Q} = P \odot Q\). Complementing one input of an XOR complements the result, so \(\bar{P} \oplus Q = P \odot Q\). Complementing both inputs cancels out, so \(\bar{P} \oplus \bar{Q} = P \oplus Q\).
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