Key Concepts and Formulas

• Converse of a Conditional Statement: The converse of a statement $p \Rightarrow q$ is $q \Rightarrow p$.
• Conditional Statement as Disjunction: $p \Rightarrow q$ is logically equivalent to $(\sim p) \vee q$.
• De Morgan's Laws: $\sim (p \wedge q) \equiv (\sim p) \vee (\sim q)$ and $\sim (p \vee q) \equiv (\sim p) \wedge (\sim q)$.

Step-by-Step Solution

Step 1: State the given conditional statement.

We are given the statement $((\sim p) \wedge q) \Rightarrow r$.

Step 2: Find the converse of the given statement.

The converse of a statement $p \Rightarrow q$ is $q \Rightarrow p$. Therefore, the converse of $((\sim p) \wedge q) \Rightarrow r$ is $r \Rightarrow ((\sim p) \wedge q)$.

Step 3: Express the converse as a disjunction.

We know that $p \Rightarrow q$ is equivalent to $(\sim p) \vee q$. Thus, $r \Rightarrow ((\sim p) \wedge q)$ is equivalent to $(\sim r) \vee ((\sim p) \wedge q)$.

Step 4: Manipulate the expression to match one of the given options.

We have $(\sim r) \vee ((\sim p) \wedge q)$. Our target is option (C), which is $(p \vee (\sim q)) \Rightarrow (\sim r)$. Let's convert this option into a disjunction: $(\sim (p \vee (\sim q))) \vee (\sim r)$.

Now let's simplify the negation using De Morgan's Law: $(\sim p \wedge \sim(\sim q)) \vee (\sim r)$, which becomes $(\sim p \wedge q) \vee (\sim r)$.

This is precisely what we have derived in Step 3: $(\sim r) \vee ((\sim p) \wedge q)$. Therefore, the converse is equivalent to $(\sim (p \vee (\sim q))) \vee (\sim r)$, which is the same as $(p \vee (\sim q)) \Rightarrow (\sim r)$.

Step 5: Write out the final form of the converse.

The converse of $((\sim p) \wedge q) \Rightarrow r$ is $r \Rightarrow ((\sim p) \wedge q)$, which is equivalent to $(\sim r) \vee ((\sim p) \wedge q)$. Further simplification gives $((\sim p) \wedge q) \vee (\sim r)$, and finally $(p \vee (\sim q)) \Rightarrow (\sim r)$.

Common Mistakes & Tips

• Confusing Converse, Inverse, and Contrapositive: Make sure to remember the definitions of each. The converse switches the hypothesis and conclusion. The inverse negates both. The contrapositive switches and negates both.
• Applying De Morgan's Laws Correctly: Be careful with the negations when applying De Morgan's laws. $\sim (p \wedge q) = (\sim p) \vee (\sim q)$ and $\sim (p \vee q) = (\sim p) \wedge (\sim q)$.
• Using $p \Rightarrow q \equiv \sim p \vee q$: This equivalence is very useful for simplifying conditional statements.

Summary

We started with the given statement $((\sim p) \wedge q) \Rightarrow r$ and found its converse, $r \Rightarrow ((\sim p) \wedge q)$. We then used the equivalence $p \Rightarrow q \equiv (\sim p) \vee q$ to express the converse as a disjunction. Applying De Morgan's Law and converting back to a conditional statement, we arrived at $(p \vee (\sim q)) \Rightarrow (\sim r)$, which corresponds to option (C).

Final Answer

The final answer is \boxed{(p \vee (\sim q)) \Rightarrow (\sim r)}, which corresponds to option (C).