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

Topic: Mathematical Logic - Conditional Statements
Step 1: Understanding the Question:
The question asks for the "converse" of a logical implication.
Logical implications follow the structure "If P, then Q".
Step 2: Key Formula or Approach:
For any conditional statement \(P \rightarrow Q\):
1. The Converse is \(Q \rightarrow P\).
2. The Inverse is \(\sim P \rightarrow \sim Q\).
3. The Contrapositive is \(\sim Q \rightarrow \sim P\).
Step 3: Detailed Explanation:
1. Identify the components of the given statement \(((\sim p) \land q) \rightarrow r\):
Hypothesis (\(P\)): \((\sim p) \land q\)
Conclusion (\(Q\)): \(r\)
2. To form the converse, swap the hypothesis and the conclusion.
3. Resulting expression: \(r \rightarrow ((\sim p) \land q)\).
Step 4: Final Answer:
The converse is \(r \rightarrow ((\sim p) \land q)\).