Question

Consider the following statements \[ P:\ \text{Suman is brilliant} \] \[ Q:\ \text{Suman is rich} \] \[ R:\ \text{Suman is honest} \] The negation of the statement “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as:

• A. $\sim(P \land \sim R) \leftrightarrow Q$
• B. $\sim P \land (Q \leftrightarrow \sim R)$
• C. $\sim(Q \leftrightarrow (P \land \sim R))$
• D. $\sim Q \leftrightarrow \sim P \land R$

Hint:

The negation of an “if and only if” statement $A \leftrightarrow B$ is $\sim(A \leftrightarrow B)$, not $A \leftrightarrow \sim B$.

Answer 1

The Correct Option is C

Step 1: Translate the given statement into symbolic form. “Suman is brilliant and dishonest if and only if Suman is rich” Dishonest $\Rightarrow \sim R$ \[ \text{Statement}:\ (P \land \sim R) \leftrightarrow Q \]
Step 2: The negation of a statement $S$ is written as $\sim S$.
Step 3: Therefore, the negation of the given statement is: \[ \sim\big((P \land \sim R) \leftrightarrow Q\big) \]
Step 4: Rewriting in the form given in the options: \[ \sim(Q \leftrightarrow (P \land \sim R)) \]
Step 5: Hence, the correct option is (C).