Question

The converse of ((~ p) ∧ q) ⇒ r is

• A. r ⇒ (p v ~q)
• B. ~r ⇒ (~p ∧ q)
• C. ~r ⇒ (p v ~q)
• D. r ⇒ (~p ∧ q)

Answer 1

The Correct Option is D

To solve this problem, we need to find the converse of the given statement. The original statement given is \(((\sim p) \wedge q) \Rightarrow r\). Let's break this down step-by-step:

Identify the components: 
The given logical statement is \(((\sim p) \wedge q) \Rightarrow r\).
- Premise or antecedent: \((\sim p) \wedge q\)
- Conclusion or consequent: \(r\)

Understanding the converse:
In logic, the converse of an implication \(A \Rightarrow B\) is \(B \Rightarrow A\).

Apply this to find the converse:
For the given statement \(((\sim p) \wedge q) \Rightarrow r\), the converse would be rearranging such that: \(r \Rightarrow (\sim p \wedge q)\)

Select the correct option:
We compare this result to the options provided:

• \(r \Rightarrow (p \vee \sim q)\)
• \(\sim r \Rightarrow (\sim p \wedge q)\)
• \(\sim r \Rightarrow (p \vee \sim q)\)
• \(r \Rightarrow (\sim p \wedge q)\)

Conclusion:
The converse of \(((\sim p) \wedge q) \Rightarrow r\) is indeed \(r \Rightarrow (\sim p \wedge q)\). Therefore, option 4 is correct.