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:
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:
Show Hint
The negation of an “if and only if” statement $A \leftrightarrow B$ is $\sim(A \leftrightarrow B)$, not $A \leftrightarrow \sim B$.
Updated On: Jan 14, 2026
- $\sim(P \land \sim R) \leftrightarrow Q$
- $\sim P \land (Q \leftrightarrow \sim R)$
- $\sim(Q \leftrightarrow (P \land \sim R))$
- $\sim Q \leftrightarrow \sim P \land R$
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
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).
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).
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