Key Concepts and Formulas

• Negation: The negation of a statement $p$ is denoted by $\sim p$. If $p$ is true, then $\sim p$ is false, and if $p$ is false, then $\sim p$ is true.
• 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:
  • $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$
  • $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$

Step-by-Step Solution

Step 1: Negate the given expression. We are given the expression $\sim s \vee (\sim r \wedge s)$ and we want to find its negation. We apply the negation operator to the entire expression: $\sim [\sim s \vee (\sim r \wedge s)]$

Step 2: Apply De Morgan's Law. Using De Morgan's Law, we distribute the negation: $\sim (\sim s) \wedge \sim (\sim r \wedge s)$

Step 3: Simplify the double negation and apply De Morgan's Law again. The negation of a negation cancels out, so $\sim (\sim s) \equiv s$. Apply De Morgan's Law to the second term: $s \wedge (\sim (\sim r) \vee \sim s)$

Step 4: Simplify the double negation. Again, the negation of a negation cancels out, so $\sim (\sim r) \equiv r$. $s \wedge (r \vee \sim s)$

Step 5: Apply the distributive law (optional, but helpful for comparison). Although not strictly necessary, we can distribute $s$ to rewrite the expression: $(s \wedge r) \vee (s \wedge \sim s)$

Step 6: Simplify using the contradiction. The expression $s \wedge \sim s$ is always false (a contradiction), so it is equivalent to $F$. $(s \wedge r) \vee F$

Step 7: Simplify using the identity law. Since $(s \wedge r) \vee F \equiv (s \wedge r)$, we have: $s \wedge (r \vee \sim s)$ Comparing the result with the options provided, we can see that option (A) is $s \vee (r \vee \sim s)$, which is not equivalent. Option (B) is $s \wedge r$, which is also not equivalent to $s \wedge (r \vee \sim s)$. However, we have derived $s \wedge (r \vee \sim s)$. We can check if $s \wedge (r \vee \sim s)$ is equivalent to $s \wedge r$. Consider $s \wedge (r \vee \sim s) \equiv (s \wedge r) \vee (s \wedge \sim s) \equiv (s \wedge r) \vee F \equiv s \wedge r$. Thus, option (B) is the correct answer.

However, the correct answer is stated as (A). Let's re-examine the solution.

We arrived at $s \wedge (r \vee \sim s)$.

Option (A) is $s \vee (r \vee \sim s)$. Option (B) is $s \wedge r$. Option (C) is $s \wedge \sim r$. Option (D) is $s \wedge (r \wedge \sim s)$.

Recall the absorption law: $p \wedge (p \vee q) \equiv p$ and $p \vee (p \wedge q) \equiv p$. Also, $p \vee (\sim p \wedge q) \equiv p \vee q$.

Let's consider the expression we found: $s \wedge (r \vee \sim s)$. We can rewrite this as $(s \wedge r) \vee (s \wedge \sim s) \equiv (s \wedge r) \vee F \equiv s \wedge r$. Thus, option (B) is correct.

Let's check option (A): $s \vee (r \vee \sim s) \equiv (s \vee \sim s) \vee r \equiv T \vee r \equiv T$. This is not the negation of the original expression.

The correct answer is (B), but we are told the correct answer is (A).

Let's reconsider the original expression: $\sim s \vee (\sim r \wedge s)$. Its negation is $\sim (\sim s \vee (\sim r \wedge s)) \equiv s \wedge \sim (\sim r \wedge s) \equiv s \wedge (r \vee \sim s) \equiv (s \wedge r) \vee (s \wedge \sim s) \equiv (s \wedge r) \vee F \equiv s \wedge r$.

Thus, the negation of the given expression is $s \wedge r$.

The given answer is (A): $s \vee (r \vee \sim s) \equiv s \vee \sim s \vee r \equiv T \vee r \equiv T$. This is always true, and therefore cannot be the negation of the original expression.

The initial solution and the answer key are incorrect. The correct answer is $s \wedge r$.

Common Mistakes & Tips

• Carefully apply De Morgan's Laws and distributive laws. A small mistake can lead to an incorrect answer.
• Remember the common logical equivalences, such as double negation, contradiction, and tautology.
• Double-check your work, especially when dealing with multiple negations and logical connectives.

Summary

We found the negation of the given expression $\sim s \vee (\sim r \wedge s)$ by applying De Morgan's Laws and simplifying. The negation simplifies to $s \wedge (r \vee \sim s)$, which is equivalent to $s \wedge r$. The provided correct answer (A) is incorrect.

Final Answer

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