Key Concepts and Formulas

• Logical Connectives: Understanding the logical connectives such as AND ($\wedge$), OR ($\vee$), NOT ($\sim$), implication ($\rightarrow$), and biconditional ($\leftrightarrow$).
• Biconditional: The biconditional statement $p \leftrightarrow q$ is true if and only if both $p$ and $q$ have the same truth value (both true or both false).
• Negation: The negation of a statement reverses its truth value.

Step-by-Step Solution

Step 1: Express "Suman is brilliant and dishonest" in symbolic form.

We are given: P : Suman is brilliant Q : Suman is rich R : Suman is honest

"Suman is brilliant and dishonest" can be written as $P \wedge \sim R$. This is because "and" corresponds to the logical conjunction $\wedge$, and "dishonest" is the negation of "honest," so we use $\sim R$.

Step 2: Express "Suman is brilliant and dishonest if and only if Suman is rich" in symbolic form.

The phrase "if and only if" corresponds to the biconditional connective, $\leftrightarrow$. Therefore, the statement "Suman is brilliant and dishonest if and only if Suman is rich" can be written as $(P \wedge \sim R) \leftrightarrow Q$.

Step 3: Find the negation of the statement $(P \wedge \sim R) \leftrightarrow Q$.

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

Step 4: Use the commutative property of the biconditional.

The biconditional connective is commutative, meaning that $p \leftrightarrow q$ is logically equivalent to $q \leftrightarrow p$. Therefore, we can rewrite $\sim [(P \wedge \sim R) \leftrightarrow Q]$ as $\sim [Q \leftrightarrow (P \wedge \sim R)]$.

Step 5: Compare the result with the given options.

We obtained $\sim [Q \leftrightarrow (P \wedge \sim R)]$, which matches option (A).

Common Mistakes & Tips

• Careful with Negation: When negating compound statements, remember to apply De Morgan's Laws or other relevant rules if necessary. In this problem, only the entire biconditional statement needs to be negated.
• Biconditional vs. Implication: Remember the difference between "if... then..." (implication, $\rightarrow$) and "if and only if" (biconditional, $\leftrightarrow$).
• Order of Operations: Pay attention to the order of operations for logical connectives. Parentheses are crucial for grouping correctly.

Summary

The problem requires translating an English statement into symbolic logic and then finding its negation. We first identified the symbolic representation of "Suman is brilliant and dishonest if and only if Suman is rich" as $(P \wedge \sim R) \leftrightarrow Q$. Then, we negated this expression to get $\sim [(P \wedge \sim R) \leftrightarrow Q]$. Using the commutative property of the biconditional, we rewrote this as $\sim [Q \leftrightarrow (P \wedge \sim R)]$, which matches option (A).

The final answer is \boxed{\sim \left[ {Q \leftrightarrow \left( {P \wedge \sim R} \right)} \right]}, which corresponds to option (A).