The converse of ((~ p) ∧ q) ⇒ r is
- r ⇒ (p v ~q)
- ~r ⇒ (~p ∧ q)
- ~r ⇒ (p v ~q)
- r ⇒ (~p ∧ q)
The Correct Option is D
Solution and Explanation
Given: The statement is \(((\sim p) \land q) \implies r\). The converse of a conditional statement \((A \implies B)\) is defined as \((B \implies A)\).
Step 1: Identify A and B.
Here:
\(A = ((\sim p) \land q), B = r.\)
Thus, the converse is:
\(r \implies ((\sim p) \land q).\)
Step 2: Express the negation of the implication.
The negation of \(((\sim p) \land q) \implies r\) is:
\[\sim(((\sim p) \land q) \implies r) = (\sim r) \implies ((\sim p) \land q).\]
Step 3: Derive the logical equivalence.
The converse can also be written equivalently as:
\[r \implies ((\sim p) \land q) \implies (\sim((\sim p) \land q)) \implies (\sim r).\]
Simplifying further:
\[(p \lor (\sim q)) \implies (\sim r).\]
Final Answer: The converse is \((p \lor (\sim q)) \implies (\sim r).\)
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