Key Concepts and Formulas

• Biconditional Equivalence: $A \Leftrightarrow B \equiv (A \Rightarrow B) \wedge (B \Rightarrow A) \equiv (\sim A \vee B) \wedge (\sim B \vee A)$
• Negation of Biconditional: $\sim (A \Leftrightarrow B) \equiv (A \wedge \sim B) \vee (B \wedge \sim A)$
• De Morgan's Laws:
  • $\sim (A \wedge B) \equiv \sim A \vee \sim B$
  • $\sim (A \vee B) \equiv \sim A \wedge \sim B$

Step-by-Step Solution

Step 1: Translate the given statement into symbolic form.

The statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be written as: $(P \wedge \sim R) \Leftrightarrow \sim Q$ We are given that $P$ represents "Ramu is intelligent," $Q$ represents "Ramu is rich," and $R$ represents "Ramu is not honest." Therefore, $\sim R$ represents "Ramu is honest," and $\sim Q$ represents "Ramu is not rich."

Step 2: Find the negation of the statement.

We need to find the negation of $(P \wedge \sim R) \Leftrightarrow \sim Q$, which is $\sim[(P \wedge \sim R) \Leftrightarrow \sim Q]$.

Step 3: Apply the negation of the biconditional equivalence.

Using the formula $\sim (A \Leftrightarrow B) \equiv (A \wedge \sim B) \vee (B \wedge \sim A)$, we have: $\sim[(P \wedge \sim R) \Leftrightarrow \sim Q] \equiv [(P \wedge \sim R) \wedge \sim (\sim Q)] \vee [\sim Q \wedge \sim (P \wedge \sim R)]$

Step 4: Simplify the expression.

Since $\sim (\sim Q) \equiv Q$, we have: $[(P \wedge \sim R) \wedge Q] \vee [\sim Q \wedge \sim (P \wedge \sim R)]$

Step 5: Apply De Morgan's Law.

Using De Morgan's Law, $\sim (P \wedge \sim R) \equiv \sim P \vee \sim (\sim R) \equiv \sim P \vee R$. Thus, the expression becomes: $[(P \wedge \sim R) \wedge Q] \vee [\sim Q \wedge (\sim P \vee R)]$

Step 6: Compare the result with the given options.

The expression we obtained is $((P \wedge \sim R) \wedge Q) \vee ((\sim Q) \wedge ((\sim P) \vee R))$. This matches option (A).

Common Mistakes & Tips

• Remember to use De Morgan's Laws correctly when negating conjunctions or disjunctions.
• Be careful with the negation of negations (e.g., $\sim (\sim Q) = Q$).
• When translating statements into symbolic form, pay close attention to the meaning of each variable and its negation.

Summary

We first translated the given statement into symbolic form as $(P \wedge \sim R) \Leftrightarrow \sim Q$. Then we found the negation of this statement using the negation of the biconditional equivalence and De Morgan's laws. After simplifying the expression, we obtained $((P \wedge \sim R) \wedge Q) \vee ((\sim Q) \wedge ((\sim P) \vee R))$, which corresponds to option (A).

Final Answer

The final answer is \boxed{((P \wedge(\sim R)) \wedge Q) \vee((\sim Q) \wedge((\sim P) \vee R))}, which corresponds to option (A).