Let \(p\) and \(q\) be any two propositions. Consider the following propositional statements.
\(S_{1} : p \to q\), \(S_{2} : \neg p \wedge q\), \(S_{3} : \neg p \vee q\), \(S_{4} : \neg p \vee \neg q\),
where \(\wedge\) denotes conjunction (AND operation), \(\vee\) denotes disjunction (OR operation), and \(\neg\) denotes negation (NOT operation). Which one of the following options is correct? (Note: \(\equiv\) denotes logical equivalence)
Answer
(A)
Explanation
\(p \to q\) is false only when \(p\) is true and \(q\) is false, and \(\neg p \vee q\) is false in exactly the same case, so \(S_{1} \equiv S_{3}\).
Every other pair differs on at least one row of the truth table:
- \(S_{2}\) and \(S_{3}\): with \(p\) and \(q\) both true, \(S_{2}\) is false but \(S_{3}\) is true.
- \(S_{2}\) and \(S_{4}\): with both false, \(S_{2}\) is false but \(S_{4}\) is true.
- \(S_{1}\) and \(S_{4}\): with both true, \(S_{1}\) is true but \(S_{4}\) is false.
Answer: A