Let ⊕ and ⊙ denote the Exclusive OR and Exclusive NOR operations,…
2018
Let ⊕ and ⊙ denote the Exclusive OR and Exclusive NOR operations, respectively. Which one of the following is NOT CORRECT
- A.
\(\overline{P \oplus Q} = P \odot Q\) - B.
\(\overline{P} \oplus Q = P \odot Q\) - C.
\(\overline{P} \oplus \overline{Q} = P \oplus Q\) - D.
\(P \oplus \overline{P} \oplus Q = ( P \odot \overline{P} \odot \overline{Q})\)
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Correct answer: D
Answer: The incorrect statement is: P ⊕ ¬P ⊕ Q = ( P ⊙ ¬P ⊙ ¬Q).
Evaluate the left-hand side:
P ⊕ ¬P ⊕ Q = (P ⊕ ¬P) ⊕ Q = 1 ⊕ Q = ¬Q.
Reason: P ⊕ ¬P is always 1 because a value differs from its negation.
Evaluate the right-hand side:
P ⊙ ¬P ⊙ ¬Q = (P ⊙ ¬P) ⊙ ¬Q = 0 ⊙ ¬Q = Q.
Reason: P ⊙ ¬P is always 0 because P and its negation are never equal; 0 ⊙ ¬Q evaluates to Q.
Conclusion: Left-hand side equals ¬Q while right-hand side equals Q, so the equality does not hold in general.
Quick confirmations of the other statements:
Overline(P ⊕ Q) = P ⊙ Q: This is the definition of XNOR (exclusive NOR) as the negation of XOR.
¬P ⊕ Q = P ⊙ Q: A truth-table check shows both expressions are true exactly when P and Q are equal.
¬P ⊕ ¬Q = P ⊕ Q: Complementing both inputs does not change whether the inputs differ, so XOR is unchanged.
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