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: $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$

Step-by-Step Solution

Step 1: Write down the given Boolean expression. We begin by stating the expression we want to negate. $E = \sim s \vee (\sim r \wedge s)$

Step 2: Negate the entire expression. We apply the negation to the given expression. $\sim E = \sim (\sim s \vee (\sim r \wedge s))$

Step 3: Apply De Morgan's Law. We use De Morgan's Law to distribute the negation across the 'or' operation. $\sim E = \sim (\sim s) \wedge \sim (\sim r \wedge s)$

Step 4: Simplify the negation of negation. We simplify $\sim(\sim s)$ to $s$. $\sim E = s \wedge \sim (\sim r \wedge s)$

Step 5: Apply De Morgan's Law again. We use De Morgan's Law to distribute the negation across the 'and' operation inside the parentheses. $\sim E = s \wedge (\sim (\sim r) \vee \sim s)$

Step 6: Simplify the negation of negation again. We simplify $\sim(\sim r)$ to $r$. $\sim E = s \wedge (r \vee \sim s)$

Step 7: Apply the distributive law. We distribute $s$ across the 'or' operation. $\sim E = (s \wedge r) \vee (s \wedge \sim s)$

Step 8: Simplify $s \wedge \sim s$. The expression $s \wedge \sim s$ is always false, which we represent as $F$. $\sim E = (s \wedge r) \vee F$

Step 9: Simplify the expression. Since anything 'or' false is itself, we have $(s \wedge r) \vee F \equiv s \wedge r$. $\sim E = s \wedge r$

Common Mistakes & Tips

• Remember to apply De Morgan's Laws correctly. Pay attention to changing 'and' to 'or' and vice versa when negating.
• Be careful with parentheses. Distribute the negation properly to all terms within the parentheses.
• Recognize that $p \wedge \sim p$ is always false (F) and $p \vee \sim p$ is always true (T).

Summary

We started with the expression $\sim s \vee (\sim r \wedge s)$ and negated it. Using De Morgan's Laws and the distributive property, we simplified the negated expression step-by-step. We arrived at the simplified expression $s \wedge r$, which represents the negation of the original expression.

Final Answer

The final answer is \boxed{s \wedge r}, which corresponds to option (D).