Key Concepts and Formulas

• De Morgan's Laws:
  • $\sim (p \wedge q) \equiv \sim p \vee \sim q$
  • $\sim (p \vee q) \equiv \sim p \wedge \sim q$
• Negation of Negation: $\sim (\sim p) \equiv p$
• Distributive Law: $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$ and $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$

Step-by-Step Solution

Step 1: Write the given statement. We are given the statement $\sim p \wedge (p \vee q)$. We want to find its negation.

Step 2: Negate the given statement. We need to find $\sim [\sim p \wedge (p \vee q)]$. This is the negation of the entire expression.

Step 3: Apply De Morgan's Law. Using De Morgan's Law, we can rewrite the negation of the conjunction as a disjunction of negations: $\sim [\sim p \wedge (p \vee q)] \equiv \sim (\sim p) \vee \sim (p \vee q)$

Step 4: Simplify the negation of negation. The negation of $\sim p$ is simply $p$. So, $\sim (\sim p) \vee \sim (p \vee q) \equiv p \vee \sim (p \vee q)$

Step 5: Apply De Morgan's Law again. Apply De Morgan's Law to the negation of the disjunction: $p \vee \sim (p \vee q) \equiv p \vee (\sim p \wedge \sim q)$

Step 6: Apply the Distributive Law. Distribute $p$ over the conjunction: $p \vee (\sim p \wedge \sim q) \equiv (p \vee \sim p) \wedge (p \vee \sim q)$

Step 7: Simplify using the law of excluded middle. We know that $p \vee \sim p$ is always true (a tautology). Representing "true" as 1, we have: $(p \vee \sim p) \wedge (p \vee \sim q) \equiv 1 \wedge (p \vee \sim q)$

Step 8: Simplify the expression. Since "true AND something" is just "something", we get: $1 \wedge (p \vee \sim q) \equiv p \vee \sim q$

Therefore, the negation of the given statement is $p \vee \sim q$.

Common Mistakes & Tips

• Remember De Morgan's Laws correctly. A common mistake is to forget to negate both parts of the conjunction or disjunction.
• Be careful with the order of operations. Negation applies to the smallest possible expression first.
• Recognize common logical equivalences, like $p \vee \sim p \equiv \text{True}$.

Summary

We started by negating the given statement. Then, we applied De Morgan's Laws twice and the negation of negation rule to simplify the expression. Finally, we used the distributive law and the law of excluded middle to arrive at the simplified negation, which is $p \vee \sim q$.

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