Question

Let \( p \) and \( q \) be any two propositions. Consider the following propositional statements.
S1: \( p \rightarrow q \), S2: \( \neg p \land q \), S3: \( \neg p \lor q \), S4: \( \neg p \lor \neg q \)
where \( \land \) denotes conjunction (AND operation), \( \lor \) denotes disjunction (OR operation), and \( \neg \) denotes negation (NOT operation).
(Note: \( \equiv \) denotes logical equivalence)
Which one of the following options is correct?

• A. \( S_1 \equiv S_3 \)
• B. \( S_2 \equiv S_3 \)
• C. \( S_2 \equiv S_4 \)
• D. \( S_1 \equiv S_4 \)

Hint:

Implications \( p \rightarrow q \) are logically equivalent to disjunctions \( \neg p \lor q \). Use truth tables to verify logical equivalences between logical operations.

Answer 1

The Correct Option is A

Analyze the statements.
S1: \( p \rightarrow q \) is logically equivalent to \( \neg p \lor q \) by the definition of implication.
S2: \( \neg p \land q \) represents a conjunction, and is not logically equivalent to the other expressions.
S3: \( \neg p \lor q \) is logically equivalent to \( p \rightarrow q \), which matches S1.
S4: \( \neg p \lor \neg q \) is different and is not logically equivalent to S1 or S3.
Thus, S1 is logically equivalent to S3, making Option (A) the correct answer.