Key Concepts and Formulas

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

Step-by-Step Solution

Step 1: Apply De Morgan's Law to negate the entire statement. We are given the statement $(p \vee q) \wedge (q \vee (\sim r))$. To find its negation, we apply De Morgan's Law: $\sim(A \wedge B) \equiv (\sim A) \vee (\sim B)$. $\sim[(p \vee q) \wedge (q \vee (\sim r))] \equiv \sim(p \vee q) \vee \sim(q \vee (\sim r))$

Step 2: Apply De Morgan's Law again to negate the individual disjunctions. Now we apply De Morgan's Law to $\sim(p \vee q)$ and $\sim(q \vee (\sim r))$. $\sim(p \vee q) \equiv (\sim p) \wedge (\sim q)$ $\sim(q \vee (\sim r)) \equiv (\sim q) \wedge \sim(\sim r) \equiv (\sim q) \wedge r$ Substituting these results back into the expression from Step 1: $(\sim p \wedge \sim q) \vee (\sim q \wedge r)$

Step 3: Apply the Distributive Law. We can factor out $\sim q$ from the expression: $(\sim p \wedge \sim q) \vee (\sim q \wedge r) \equiv \sim q \wedge (\sim p \vee r)$

Step 4: Apply the Commutative Law. Using the commutative law for conjunction: $\sim q \wedge (\sim p \vee r) \equiv (\sim p \vee r) \wedge \sim q$

Common Mistakes & Tips

• Remember to apply De Morgan's Laws carefully, negating each component and changing $\wedge$ to $\vee$ and vice versa.
• Pay close attention to the negation of negations: $\sim(\sim r) = r$.
• Factoring out common terms using distributive laws can simplify the expression significantly.

Summary

We started by applying De Morgan's Laws to negate the given statement. Then, we simplified the resulting expression by applying De Morgan's Laws again and using the distributive property to factor out a common term. Finally, using commutative law we rearranged the terms to arrive at the final simplified form of the negation.

Final Answer

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