GATE 2025 DA – Question 15
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)
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Show answer and explanation
Correct answer: (A) $S_1 \equiv S_3$
Explanation
The implication $p \to q$ is the same as $\neg p \vee q$, so $S_1 \equiv S_3$. $S_2$ is a conjunction, which is true only when $p$ is false and $q$ is true, and that differs from $S_3$ (true also when $p$ and $q$ are both true) and from $S_4$ (true also when $q$ is false).