Key Concepts and Formulas

• Implication: $p \Rightarrow q \equiv \sim p \vee q$
• De Morgan's Laws:
  • $\sim (p \wedge q) \equiv \sim p \vee \sim q$
  • $\sim (p \vee q) \equiv \sim p \wedge \sim q$
• Distributive Law: $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$
• Tautology: $p \vee \sim p \equiv T$ (where T represents True)
• Identity Law: $p \wedge T \equiv p$

Step-by-Step Solution

Step 1: Rewrite the implication using the equivalence $p \Rightarrow q \equiv \sim p \vee q$. We have $\sim (p \wedge (p \Rightarrow \sim q)) = \sim (p \wedge (\sim p \vee \sim q))$ We are replacing $p \Rightarrow \sim q$ with $\sim p \vee \sim q$.

Step 2: Apply De Morgan's Law to the outer negation. $\sim (p \wedge (\sim p \vee \sim q)) = \sim p \vee \sim (\sim p \vee \sim q)$ We are using $\sim (p \wedge q) = \sim p \vee \sim q$, where $q$ is $(\sim p \vee \sim q)$.

Step 3: Apply De Morgan's Law again to the inner negation. $\sim p \vee \sim (\sim p \vee \sim q) = \sim p \vee (p \wedge q)$ We are using $\sim (p \vee q) = \sim p \wedge \sim q$, where $p$ is $\sim p$ and $q$ is $\sim q$. Thus $\sim(\sim p \vee \sim q) = \sim(\sim p) \wedge \sim(\sim q) = p \wedge q$.

Step 4: Apply the distributive law. $\sim p \vee (p \wedge q) = (\sim p \vee p) \wedge (\sim p \vee q)$ We are using $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$, where $a$ is $\sim p$, $b$ is $p$, and $c$ is $q$.

Step 5: Simplify using the tautology $p \vee \sim p \equiv T$. $(\sim p \vee p) \wedge (\sim p \vee q) = T \wedge (\sim p \vee q)$ Since $\sim p \vee p$ is always true, we replace it with $T$.

Step 6: Simplify using the identity law $p \wedge T \equiv p$. $T \wedge (\sim p \vee q) = \sim p \vee q$ Since $T \wedge (\sim p \vee q)$ is logically equivalent to $\sim p \vee q$.

Common Mistakes & Tips

• Be careful when applying De Morgan's Laws. Remember to negate all terms inside the parentheses and change the operator.
• When simplifying logical expressions, it's often helpful to rewrite implications using their equivalent forms with negations and disjunctions.
• Use the distributive law carefully. Ensure you understand how it works with both conjunctions and disjunctions.

Summary

We started with the expression $\sim (p \wedge (p \Rightarrow \sim q))$ and simplified it using logical equivalences. We first replaced the implication with its equivalent form using disjunctions and negations. Then, we applied De Morgan's Laws and the distributive law to arrive at the simplified expression $\sim p \vee q$. This expression is logically equivalent to the original expression.

Final Answer The final answer is $\boxed{\sim p \vee q}$, which corresponds to option (A).