Consider the following statement.
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. `~p ^^ (q hArr ~r)`
B. `~(q hArr (p ^^ ~r)`
C. `~q hArr ~p ^^ r`
D. `~(p ^^ ~r) hArr q`

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: Identify the components of the statement
- Let \( p \): Suman is brilliant
- Let \( q \): Suman is rich
- Let \( r \): Suman is honest

The statement can be broken down as follows:
- "Suman is brilliant and dishonest" can be expressed as \( p \land \neg r \) (where \( \neg r \) means Suman is not honest).
- The entire statement can be expressed as \( (p \land \neg r) \iff q \).

### Step 2: Write the statement in logical form
The statement "Suman is brilliant and dishonest if and only if Suman is rich" can be represented logically as:
\[ (p \land \neg r) \iff q \]

### Step 3: Apply the negation
To negate the biconditional statement \( A \iff B \), we use the equivalence:
\[ \neg(A \iff B) \equiv A \land \neg B \lor \neg A \land B \]
In our case, \( A \) is \( (p \land \neg r) \) and \( B \) is \( q \).

Thus, the negation becomes:
\[ \neg((p \land \neg r) \iff q) \equiv (p \land \neg r) \land \neg q \lor \neg(p \land \neg r) \land q \]

### Step 4: Simplify the expression
Now, we can simplify \( \neg(p \land \neg r) \):
\[ \neg(p \land \neg r) \equiv \neg p \lor r \]
So, the negation can be expressed as:
\[ ((p \land \neg r) \land \neg q) \lor ((\neg p \lor r) \land q) \]

### Final Expression
The negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich" can thus be expressed as:
\[ ((p \land \neg r) \land \neg q) \lor ((\neg p \lor r) \land q) \]