Which one of the following is NOT equivalent to \(p ↔ q\)?
2015
Which one of the following is NOT equivalent to \(p ↔ q\)?
- A.
\((\neg p \lor q) \land (p \lor \neg q)\) - B.
\((\neg p \lor q) \land (q \rightarrow p)\) - C.
\((\neg p \land q) \lor (p \land \neg q)\) - D.
\((\neg p \land \neg q) \lor (p \land q)\)
Attempted by 134 students.
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Correct answer: C
Key idea: p ↔ q is true exactly when p and q have the same truth value.
Write the biconditional as implications: p ↔ q ≡ (p → q) ∧ (q → p).
Replace implications with disjunctions: p → q is (¬p ∨ q) and q → p is (¬q ∨ p), so p ↔ q ≡ (¬p ∨ q) ∧ (¬q ∨ p).
Another equivalent form is (p ∧ q) ∨ (¬p ∧ ¬q), which directly expresses that p and q are both true or both false.
Compare each given formula to these forms: the expression (¬p ∧ q) ∨ (p ∧ ¬q) is the exclusive-or (true when p and q differ), so it is not equivalent to p ↔ q. All other listed formulas match one of the equivalent forms above and therefore are equivalent to p ↔ q.
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