Key Concepts and Formulas

• Implication: $p \to q \equiv \sim p \vee q$
• De Morgan's Law: $\sim (p \wedge q) \equiv \sim p \vee \sim q$
• Associative Law: $p \vee (q \vee r) \equiv (p \vee q) \vee r$

Step-by-Step Solution

Step 1: Rewrite the implication using the implication equivalence. We want to simplify $p \to \sim (p \wedge \sim q)$. Using the implication rule $p \to q \equiv \sim p \vee q$, we can rewrite the given expression as: $p \to \sim (p \wedge \sim q) \equiv \sim p \vee \sim (p \wedge \sim q)$ The reason for doing this is to get rid of the implication and work with conjunctions, disjunctions, and negations, which are easier to manipulate.

Step 2: Apply De Morgan's Law. Now, we apply De Morgan's Law to the term $\sim (p \wedge \sim q)$. De Morgan's Law states that $\sim (p \wedge q) \equiv \sim p \vee \sim q$. Therefore: $\sim (p \wedge \sim q) \equiv \sim p \vee \sim (\sim q) \equiv \sim p \vee q$ Substituting this back into the expression from Step 1, we get: $\sim p \vee \sim (p \wedge \sim q) \equiv \sim p \vee (\sim p \vee q)$ The reason for this step is to further simplify the expression by removing the conjunction within the negation.

Step 3: Apply the Associative Law. Since the expression now only contains disjunctions, we can apply the associative law, which states that $p \vee (q \vee r) \equiv (p \vee q) \vee r$. In our case, this means: $\sim p \vee (\sim p \vee q) \equiv (\sim p \vee \sim p) \vee q$ The associative law allows us to regroup the terms without changing the meaning of the expression.

Step 4: Simplify the expression using the idempotent law. The expression $(\sim p \vee \sim p)$ is equivalent to $\sim p$ because $p \vee p \equiv p$. Therefore: $(\sim p \vee \sim p) \vee q \equiv \sim p \vee q$ This simplification is crucial to arriving at the final answer.

Common Mistakes & Tips

• Remember the implication rule: $p \to q \equiv \sim p \vee q$. This is fundamental to solving these types of problems.
• Be careful when applying De Morgan's Law. Remember to negate each term inside the parentheses and change the conjunction to a disjunction (or vice versa).
• Don't forget the associative and commutative properties of logical operators, as they can simplify the expression.

Summary

We started with the expression $p \to \sim (p \wedge \sim q)$ and used the implication rule to rewrite it as $\sim p \vee \sim (p \wedge \sim q)$. Then we applied De Morgan's Law to simplify the negated conjunction, resulting in $\sim p \vee (\sim p \vee q)$. Finally, using the associative and idempotent laws, we simplified the expression to $\sim p \vee q$.

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