Negation of ((p q) (q p)) is: — Solution | JEE Mathematics

MCQMediumJEE 2023Sets & Operations

Negation of $((p \to q) \to (q \to p))$ is:

Answer

Correct answer:D

Standard Method

Given: The logical expression is $((p \to q) \to (q \to p))$ .

Find: Its negation.

Use the implication identity $A \to B \equiv (\sim A) \lor B$ and the negation identity $\sim(A \to B) \equiv A \land (\sim B)$ .

From the solution:

(p \to q) \to (q \to p) \equiv \sim(p \to q) \lor (q \to p)

Also,

p \to q \equiv (\sim p) \lor q, \qquad q \to p \equiv (\sim q) \lor p

Step-by-step simplification

Now negate the implication directly:

\sim\big((p \to q) \to (q \to p)\big) \equiv (p \to q) \land \sim(q \to p)

Substitute the equivalent forms:

(p \to q) \land \sim(q \to p) = \big((\sim p) \lor q\big) \land \sim\big((\sim q) \lor p\big)

Apply De Morgan's law:

\sim\big((\sim q) \lor p\big) = q \land (\sim p)

So,

\big((\sim p) \lor q\big) \land \big(q \land (\sim p)\big)

Using absorption, this simplifies to:

q \land (\sim p)

Therefore, the correct option is D.

Shortcut using implication negation

Write

\sim\big((p \to q) \to (q \to p)\big) = (p \to q) \land \sim(q \to p)

Now,

\sim(q \to p) = q \land (\sim p)

Since $q \land (\sim p)$ already makes $p \to q$ true, the whole expression reduces immediately to:

q \land (\sim p)

Hence the correct option is D.

Using $\sim(A \to B)$ as $(\sim A) \to (\sim B)$ . This is incorrect because implication does not negate termwise. Use $\sim(A \to B) \equiv A \land (\sim B)$ instead.
Neglecting brackets in $((p \to q) \to (q \to p))$ . This changes the logical structure of the statement. Always identify the outer implication before applying negation.
Stopping at an intermediate form such as $((\sim p) \lor q) \land (q \land (\sim p))$ without simplifying. This expression further reduces by absorption to $q \land (\sim p)$ .

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