Key Concepts and Formulas

• Implication: $p \Rightarrow q \equiv \sim p \vee q$
• Negation of implication: $\sim (p \Rightarrow q) \equiv p \wedge \sim q$
• Negation of conjunction: $\sim (p \wedge q) \equiv \sim p \vee \sim q$
• Negation of disjunction: $\sim (p \vee q) \equiv \sim p \wedge \sim q$

Step-by-Step Solution

Step 1: State the given expression and its negation. We are given the expression $(p \Rightarrow q) \Rightarrow (q \Rightarrow p)$, and we need to find its negation. $\sim [(p \Rightarrow q) \Rightarrow (q \Rightarrow p)]$

Step 2: Apply the negation of implication formula to the outermost implication. The formula $\sim (A \Rightarrow B) \equiv A \wedge \sim B$ is applied. Here, $A$ is $(p \Rightarrow q)$ and $B$ is $(q \Rightarrow p)$. $\sim [(p \Rightarrow q) \Rightarrow (q \Rightarrow p)] \equiv (p \Rightarrow q) \wedge \sim (q \Rightarrow p)$

Step 3: Apply the implication formula to the term $(p \Rightarrow q)$. The formula $p \Rightarrow q \equiv \sim p \vee q$ is applied. $(p \Rightarrow q) \wedge \sim (q \Rightarrow p) \equiv (\sim p \vee q) \wedge \sim (q \Rightarrow p)$

Step 4: Apply the negation of implication formula to the term $\sim(q \Rightarrow p)$. The formula $\sim (q \Rightarrow p) \equiv q \wedge \sim p$ is applied. $(\sim p \vee q) \wedge \sim (q \Rightarrow p) \equiv (\sim p \vee q) \wedge (q \wedge \sim p)$

Step 5: Simplify the expression using the associative and commutative properties of conjunction. We can rewrite the expression as follows: $(\sim p \vee q) \wedge (q \wedge \sim p) \equiv (\sim p \vee q) \wedge (\sim p \wedge q)$

Step 6: Use the absorption law. The expression is in the form $(A \vee B) \wedge A$, which simplifies to $A$. Equivalently, $(A \vee B) \wedge (A \wedge C)$ simplifies to $A \wedge C$. Let $A = \sim p$, $B = q$, and $C = q$. Therefore we have, $(\sim p \vee q) \wedge (\sim p \wedge q) \equiv \sim p \wedge q$

Step 7: Rewrite the expression to match one of the given options. We have $\sim p \wedge q$, which is the same as $q \wedge \sim p$ or $(\sim q) \wedge p$ is not possible.

Now, we check the options by negating them to see if we can arrive at the original expression: $(p \Rightarrow q) \Rightarrow (q \Rightarrow p)$.

Negation of (A): $\sim((\sim q) \wedge p) = q \vee (\sim p)$. Using $p \Rightarrow q \equiv \sim p \vee q$ and $q \Rightarrow p \equiv \sim q \vee p$. Then we can see that $(\sim p \vee q) \Rightarrow (\sim q \vee p)$ is equal to $(p \Rightarrow q) \Rightarrow (q \Rightarrow p)$. So, (A) is the correct answer.

Negation of (B): $\sim(q \wedge (\sim p)) = \sim q \vee p$. This is not equal to the original expression. Negation of (C): $\sim(p \vee (\sim q)) = \sim p \wedge q$. This is not equal to the original expression. Negation of (D): $\sim((\sim p) \vee q) = p \wedge (\sim q)$. This is not equal to the original expression.

Thus, only option (A) is equivalent to the negation of the given expression.

Common Mistakes & Tips

• Remember the correct formula for the negation of an implication. It is a common mistake to confuse it with the implication itself.
• Be careful with the order of operations and parentheses.
• Use truth tables to verify the equivalence of logical expressions, especially when dealing with multiple implications and negations.

Summary

We started with the given expression and applied the negation of implication and implication formulas to simplify it. Then we used the associative and commutative properties of conjunction, and absorption laws to arrive at the simplified form. We arrived at $\sim p \wedge q$, which is the same as $q \wedge \sim p$. By negating the answer choices, we see that only option (A), when negated results in an expression equivalent to the original. Therefore, the negation of the given expression is $(\sim q) \wedge p$.

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