Key Concepts and Formulas

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

Step-by-Step Solution

Step 1: Simplify the given expression using the distributive law. The given expression is $q \vee ((\sim q) \wedge p)$. We can rewrite this using the distributive law as $(q \vee (\sim q)) \wedge (q \vee p)$. Why: Applying the distributive law helps to simplify the expression and potentially eliminate terms.

Step 2: Simplify further using the property $p \vee (\sim p) \equiv T$ (Tautology). Since $q \vee (\sim q)$ is always true, we can replace it with $T$. So, the expression becomes $T \wedge (q \vee p)$. Why: Recognizing tautologies simplifies the expression, as anything ANDed with a tautology is just the other thing.

Step 3: Simplify using the identity $T \wedge p \equiv p$. Since $T \wedge (q \vee p)$ is equivalent to $q \vee p$, the expression simplifies to $q \vee p$. Why: Anything ANDed with True is itself.

Step 4: Find the negation of the simplified expression. We need to find the negation of $q \vee p$, which is $\sim (q \vee p)$. Why: The question asks for the negation of the original expression, so we must negate the simplified form.

Step 5: Apply De Morgan's Law. Using De Morgan's Law, $\sim (q \vee p) \equiv (\sim q) \wedge (\sim p)$. Why: De Morgan's Law provides a direct way to negate a disjunction.

Step 6: Rewrite the expression to match the options. The negation of the expression is $(\sim q) \wedge (\sim p)$, which is equivalent to $(\sim p) \wedge (\sim q)$. Why: To match the correct option.

Common Mistakes & Tips

• Remember De Morgan's Laws correctly. A common mistake is to forget to negate both terms and change the operator.
• Always simplify the expression before negating it. This often makes the negation process easier.
• Be careful with the order of operations (AND before OR).

Summary

We started with the given expression $q \vee ((\sim q) \wedge p)$, simplified it using the distributive law and properties of logical connectives to get $q \vee p$. Then, we found the negation of this simplified expression using De Morgan's Law, which gave us $(\sim q) \wedge (\sim p)$, equivalent to $(\sim p) \wedge (\sim q)$. Thus, the negation of the given expression is $(\sim p) \wedge (\sim q)$.

Final Answer

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