Mathematical Reasoning * — JEE Main · 2022 · 25 Jul 2022 · Shift 2 Question with Solution

Consider the following statements :
P : Ramu is intelligent
Q : Ramu is rich
R : Ramu is not honest
The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as :

(A) $((P \land (\sim R)) \land Q) \land ((\sim Q) \land ((\sim P) \lor R))$
(B) $((P \land R) \land Q) \lor ((\sim Q) \land ((\sim P) \lor (\sim R)))$
(C) $((P \land R) \land Q) \land ((\sim Q) \land ((\sim P) \lor (\sim R)))$
(D) $((P \land (\sim R)) \land Q) \lor ((\sim Q) \land ((\sim P) \lor R))$ ✓ Correct

Answer: (D)

Solution

The statement is: "Ramu is intelligent and honest if and only if Ramu is not rich".

"Intelligent and honest" =

P \land (\sim R)

(since R is "not honest",

\sim R

is "honest").

"Not rich" =

\sim Q

.

The statement is:

(P \land (\sim R)) \leftrightarrow (\sim Q)

.

Negation of

A \leftrightarrow B

(A \land (\sim B)) \lor ((\sim A) \land B)

.

So negation is:

((P \land (\sim R)) \land Q) \lor ((\sim (P \land (\sim R))) \land (\sim Q))

.

Simplifying

\sim (P \land (\sim R)) = (\sim P) \lor R

.

Thus:

((P \land (\sim R)) \land Q) \lor (((\sim P) \lor R) \land (\sim Q))

, which matches option (D).

JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Single correct option

JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Mathematical Reasoning *

Solution

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