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 expressed as

Options

A. (a) `~(Q harr(P wedge~R))`
B. (b) `~Qharr~PveeR`
C. (c) `~(P wedge~R)harrQ`
D. (d) `~P wedge(Q harr~R)`

Answer

To find the negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich," we can follow these steps:

### Step 1: Understand the Original Statement
The original statement can be broken down into parts:
- Let \( P \): Suman is brilliant.
- Let \( Q \): Suman is rich.
- Let \( R \): Suman is honest.

The statement "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as:
\[ (P \land \neg R) \iff Q \]
where \( \neg R \) represents "Suman is dishonest."

### Step 2: Write the Negation
The negation of a biconditional statement \( A \iff B \) is given by:
\[ \neg(A \iff B) \equiv A \land \neg B \lor \neg A \land B \]

Applying this to our statement:
\[ \neg((P \land \neg R) \iff Q) \]
Using the biconditional negation formula, we have:
\[ (P \land \neg R) \land \neg Q \lor \neg(P \land \neg R) \land Q \]

### Step 3: Simplify the Negation
Now we can simplify each part:
1. \( \neg(P \land \neg R) \) can be simplified using De Morgan's laws:
\[ \neg(P \land \neg R) \equiv \neg P \lor R \]

Thus, we can rewrite our negation:
\[ (P \land \neg R) \land \neg Q \lor (\neg P \lor R) \land Q \]

### Step 4: Final Expression
The final expression for the negation of the original statement is:
\[ (P \land \neg R \land \neg Q) \lor (\neg P \land Q) \lor (R \land Q) \]

### Conclusion
This expression represents the negation of the original statement.