Question

The converse of the statement \( ((\sim p) \land q) \rightarrow r \) is:

• A. \(r \rightarrow ((\sim p) \land q)\)
• B. \(((\sim p) \land q) \rightarrow (\sim r)\)
• C. \(r \rightarrow (\sim p)\)
• D. \(q \rightarrow r\)

Hint:

For a conditional statement \(P \rightarrow Q\):
• Converse: \(Q \rightarrow P\)
• Inverse: \((\sim P) \rightarrow (\sim Q)\)
• Contrapositive: \((\sim Q) \rightarrow (\sim P)\) Contrapositive is always logically equivalent to the original statement.

Answer 1

The Correct Option is A

Concept: In propositional logic, the converse of a conditional statement is obtained by interchanging the hypothesis and the conclusion. If the given statement is \[ P \rightarrow Q \] then the converse is \[ Q \rightarrow P \]

Step 1: Identify hypothesis and conclusion. Given statement: \[ ((\sim p) \land q) \rightarrow r \] Here:
• Hypothesis: \( ((\sim p) \land q) \)
• Conclusion: \( r \)

Step 2: Form the converse statement. Interchanging hypothesis and conclusion: \[ r \rightarrow ((\sim p) \land q) \]

Step 3: Final answer. Thus, the converse of the given statement is \[ \boxed{r \rightarrow ((\sim p) \land q)} \]