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: $\sim (\sim p) \equiv p$
• Identity Laws:
  • $p \wedge T \equiv p$ where T represents "True"
  • $p \vee F \equiv p$ where F represents "False"
• Distributive Laws:
  • $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$
  • $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$

Step-by-Step Solution

Step 1: Find the negation of the given expression. We are given the expression $(p \wedge (\sim q)) \vee (\sim p)$. We want to find its negation, which is $\sim[(p \wedge (\sim q)) \vee (\sim p)]$.

Step 2: Apply De Morgan's Law to the outer negation. Using De Morgan's Law, we can rewrite the negation of the expression as follows: $\sim[(p \wedge (\sim q)) \vee (\sim p)] \equiv \sim(p \wedge (\sim q)) \wedge \sim(\sim p)$

Step 3: Apply De Morgan's Law to the first part of the expression. Also simplify the second part. Applying De Morgan's Law to $\sim(p \wedge (\sim q))$ gives us $(\sim p) \vee (\sim (\sim q))$. Also, $\sim(\sim p)$ simplifies to $p$. Thus, the expression becomes: $[(\sim p) \vee (\sim (\sim q))] \wedge p \equiv [(\sim p) \vee q] \wedge p$

Step 4: Apply the distributive law. Using the distributive law, we can rewrite the expression as: $[(\sim p) \vee q] \wedge p \equiv ((\sim p) \wedge p) \vee (q \wedge p)$

Step 5: Simplify the expression further. We know that $(\sim p) \wedge p$ is always false (F). Therefore, the expression simplifies to: $((\sim p) \wedge p) \vee (q \wedge p) \equiv F \vee (q \wedge p)$

Step 6: Apply the Identity Law. Since $F \vee (q \wedge p) \equiv (q \wedge p)$, the expression simplifies to: $F \vee (q \wedge p) \equiv (q \wedge p) \equiv (p \wedge q)$

Common Mistakes & Tips

• Remember to apply De Morgan's Laws correctly, paying close attention to changing AND to OR and vice versa.
• When negating a negated term (like $\sim (\sim p)$), remember it simplifies back to the original term ($p$).
• Be careful when applying distributive laws; ensure you distribute correctly over AND and OR.
• It is crucial to remember the basic logical equivalences and identities to simplify expressions effectively.

Summary

We started with the negation of the given expression and applied De Morgan's laws and the distributive property to simplify it. By identifying and using logical equivalences such as $p \wedge (\sim p) \equiv F$ and $F \vee p \equiv p$, we were able to reduce the expression to its simplest form, which is $p \wedge q$.

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