Question

The converse of \( ((\sim p) \land q) \Rightarrow r \) is:

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

Hint:

The converse of \( A \Rightarrow B \) is \( B \Rightarrow A \). Ensure to use logical equivalence rules for simplification.

Answer 1

The Correct Option is C

The converse of an implication statement \( A \Rightarrow B \) is \( B \Rightarrow A \).
For the statement \( ((\sim p) \land q) \Rightarrow r \), its converse is \( r \Rightarrow ((\sim p) \land q) \).
Rewriting \( r \Rightarrow ((\sim p) \land q) \) using logical equivalence yields \[\sim r \Rightarrow (\sim (\sim p) \lor (\sim q)) \Rightarrow (p \lor (\sim q)).\] This simplifies to \( (p \lor (\sim q)) \Rightarrow (\sim r) \).
Conclusion: The converse is \( (p \lor (\sim q)) \Rightarrow (\sim r) \).