Key Concepts and Formulas

• Implication: $p \Rightarrow q \equiv \sim p \vee q$
• De Morgan's Laws: $\sim (p \vee q) \equiv \sim p \wedge \sim q$ and $\sim (p \wedge q) \equiv \sim p \vee \sim q$
• Double Negation: $\sim(\sim p) \equiv p$

Step-by-Step Solution

Step 1: Rewrite the implication using the equivalence $p \Rightarrow q \equiv \sim p \vee q$. We are given the expression $\sim (p \Rightarrow (\sim q))$. We want to simplify this. First, we apply the implication equivalence to the expression inside the parentheses: $\sim (p \Rightarrow (\sim q)) \equiv \sim (\sim p \vee \sim q)$ This replaces the implication with an equivalent expression using negation and disjunction.

Step 2: Apply De Morgan's Law. Now we apply De Morgan's Law to the expression inside the outer negation: $\sim (\sim p \vee \sim q)$. De Morgan's Law states that $\sim (p \vee q) \equiv \sim p \wedge \sim q$. Thus, we have: $\sim (\sim p \vee \sim q) \equiv \sim (\sim p) \wedge \sim (\sim q)$ This step transforms the disjunction into a conjunction of negations.

Step 3: Simplify using Double Negation. We apply the double negation rule, $\sim (\sim p) \equiv p$, to both terms: $\sim (\sim p) \wedge \sim (\sim q) \equiv p \wedge q$ This simplifies the expression to its final form.

Common Mistakes & Tips

• Remember to apply De Morgan's Laws correctly. It's a common mistake to only negate one of the terms inside the parentheses.
• Be careful with the order of operations. Negation applies before conjunction and disjunction.
• When simplifying Boolean expressions, it is often helpful to rewrite implications using the equivalence $p \Rightarrow q \equiv \sim p \vee q$.

Summary

We started with the expression $\sim (p \Rightarrow (\sim q))$. By first rewriting the implication using the equivalence $p \Rightarrow q \equiv \sim p \vee q$, and then applying De Morgan's Law and the double negation rule, we simplified the expression to $p \wedge q$. This is the final simplified form.

Final Answer

The final answer is \boxed{p \wedge q}, which corresponds to option (A).