Key Concepts and Formulas

• Negation of an implication: $\sim (p \Rightarrow q) \equiv p \wedge \sim q$
• De Morgan's Laws:
  • $\sim (p \vee q) \equiv \sim p \wedge \sim q$
  • $\sim (p \wedge q) \equiv \sim p \vee \sim q$

Step-by-Step Solution

Step 1: State the given Boolean statement and indicate that we need to find its negation. We are given the statement $(p \vee q) \Rightarrow ((\sim r) \vee p)$ and we need to find its negation.

Step 2: Apply the negation to the entire statement. $\sim ((p \vee q) \Rightarrow ((\sim r) \vee p))$

Step 3: Use the negation of implication formula: $\sim (A \Rightarrow B) \equiv A \wedge \sim B$. Here, $A = (p \vee q)$ and $B = ((\sim r) \vee p)$. Applying the formula, we get: $(p \vee q) \wedge \sim ((\sim r) \vee p)$

Step 4: Apply De Morgan's Law to negate the second part of the expression. We have $\sim ((\sim r) \vee p) \equiv \sim (\sim r) \wedge \sim p \equiv r \wedge \sim p$. So the expression becomes: $(p \vee q) \wedge (r \wedge \sim p)$

Step 5: Distribute the conjunction $(r \wedge \sim p)$ over the disjunction $(p \vee q)$. $(p \vee q) \wedge (r \wedge \sim p) \equiv (p \wedge (r \wedge \sim p)) \vee (q \wedge (r \wedge \sim p))$

Step 6: Use the associative property of conjunction. $(p \wedge r \wedge \sim p) \vee (q \wedge r \wedge \sim p)$

Step 7: Simplify the first part of the disjunction. Since $p \wedge \sim p$ is always false (contradiction), $p \wedge r \wedge \sim p \equiv (p \wedge \sim p) \wedge r \equiv F \wedge r \equiv F$, where $F$ represents the boolean value False. So, the expression becomes: $F \vee (q \wedge r \wedge \sim p)$

Step 8: Simplify the entire expression. Since $F \vee X \equiv X$, we have: $q \wedge r \wedge \sim p$ Which can be written as $\sim p \wedge q \wedge r$

Step 9: Compare the expression to the provided options. The options are: (A) $p \wedge (\sim q) \wedge r$ (B) $(\sim p) \wedge (\sim q) \wedge r$ (C) $(\sim p) \wedge q \wedge r$ (D) $p \wedge q \wedge (\sim r)$

Our result is $\sim p \wedge q \wedge r$, which matches option (C). However, the ground truth is option (A). Let's re-examine the steps to see if we made an error.

Step 1: (p ∨ q) ⇒ ((∼ r) ∨ p) Step 2: ∼ ((p ∨ q) ⇒ ((∼ r) ∨ p)) Step 3: (p ∨ q) ∧ ∼ ((∼ r) ∨ p) Step 4: (p ∨ q) ∧ (r ∧ ∼ p) Step 5: (p ∧ r ∧ ∼ p) ∨ (q ∧ r ∧ ∼ p) Step 6: F ∨ (q ∧ r ∧ ∼ p) Step 7: (q ∧ r ∧ ∼ p)

The result is (~p ∧ q ∧ r). The "Correct Answer" is stated as (A) p ∧ (∼ q) ∧ r. There is an error in the stated "Correct Answer".

However, if we assume the question intended the answer to be (A) $p \wedge \sim q \wedge r$, we can work backwards.

If the answer is $p \wedge \sim q \wedge r$, then the original statement's negation is equivalent to that. Then the original statement is:

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

$(p \vee q) \Rightarrow ((\sim r) \vee p) \equiv \sim (p \vee q) \vee ((\sim r) \vee p) \equiv (\sim p \wedge \sim q) \vee (\sim r \vee p) \equiv \sim p \wedge \sim q \vee \sim r \vee p$

This is where the stated ground truth differs. Let's continue with our original solution.

Common Mistakes & Tips

• Remember the correct negation of an implication. It's a common source of error.
• Be careful with De Morgan's Laws and remember to apply them correctly.
• Always double-check your work, especially when dealing with multiple logical operations.

Summary

We started by negating the given Boolean statement using the negation of implication and De Morgan's Laws. We simplified the expression by distributing the conjunction and using the fact that $p \wedge \sim p$ is always false. We arrived at $\sim p \wedge q \wedge r$, which corresponds to option (C). However, according to the question, option (A) is the ground truth. This suggests an error in the question's provided "Correct Answer". Based on our derivation, the correct answer is ($\sim p) \wedge q \wedge r$.

Final Answer

The final answer is \boxed{(\sim p) \wedge q \wedge r}, which corresponds to option (C). The provided "Correct Answer" (A) is incorrect.