Consider the following statements:
P: Suman is brilliant
Q: Suman is rich
R: Suman is honest

The negation of the statement, "Suman is brilliant and dishonest if and only if Suman is rich" can be equivalently expressed as

We are given three statements:

• P: Suman is brilliant
• Q: Suman is rich
• R: Suman is honest

The statement to negate is: "Suman is brilliant and dishonest if and only if Suman is rich".

First, note that "dishonest" means not honest, so it is represented as $$\sim R$$. Therefore, "Suman is brilliant and dishonest" is written as $$P \wedge \sim R$$.

The entire statement "Suman is brilliant and dishonest if and only if Suman is rich" is logically expressed as $$(P \wedge \sim R) \leftrightarrow Q$$.

We need to find the negation of this statement: $$\sim \left[ (P \wedge \sim R) \leftrightarrow Q \right]$$.

Recall that the negation of a biconditional $$A \leftrightarrow B$$ is equivalent to $$A \leftrightarrow \sim B$$. Applying this here, let $$A = P \wedge \sim R$$ and $$B = Q$$. Then:

$$\sim \left[ (P \wedge \sim R) \leftrightarrow Q \right] \equiv (P \wedge \sim R) \leftrightarrow \sim Q$$

Since the biconditional is commutative (i.e., $$X \leftrightarrow Y$$ is the same as $$Y \leftrightarrow X$$), we can rewrite this as:

$$\sim Q \leftrightarrow (P \wedge \sim R)$$

Now, comparing this with the given options:

• Option A: $$\sim Q \leftrightarrow \sim P \vee R$$
• Option B: $$\sim Q \leftrightarrow P \vee \sim R$$
• Option C: $$\sim Q \leftrightarrow P \wedge \sim R$$
• Option D: $$\sim Q \leftrightarrow \sim P \wedge R$$

Our expression $$\sim Q \leftrightarrow (P \wedge \sim R)$$ matches Option C exactly.

Therefore, the negation of the given statement is equivalently expressed as Option C.

Hence, the correct answer is Option C.