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

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 \((p \lor (\sim q)) \implies (\sim r).\)