A stem like "Which of the following is the negation of 'If A, then B'?" appears every year in JEE and board logic sections, and every year the trap options are the same:

The correct answer is (D) A \land \lnot B. Not any of the other three "if...then" options, no matter how tempting they look. The negation of an implication is not another implication — it is a conjunction.

The underlying identity is:

\lnot(A \Rightarrow B) \equiv A \land \lnot B.

That is one of the most tested equivalences in propositional logic. Install the rewrite as a reflex and this entire question class becomes a five-second single-line calculation.

The recognition cue

Any of these stems should trigger the reflex:

Every one of these is asking for A \land \lnot B.

The two-line derivation

If you ever forget the rewrite, it is two lines from the fundamental equivalence of implication:

Line 1 (implication as disjunction): A \Rightarrow B \equiv \lnot A \lor B.

Line 2 (negate, apply De Morgan): \lnot(A \Rightarrow B) \equiv \lnot(\lnot A \lor B) \equiv A \land \lnot B.

See Logic and Propositions for the full truth-table proof and De Morgan for negating compound statements for the distribution step.

Why this is the only possible negation: an implication A \Rightarrow B fails in exactly one row of its four-row truth table — the row with A = T and B = F. The negation must be T in exactly that row. The expression A \land \lnot B is T in exactly that row — and F in the other three. So it matches the negation row for row. Nothing else does.

Why each "if...then" trap option is wrong

(A) Inverse: "If \lnot A, then \lnot B"

Its truth table is T, T, F, T — different from the negation pattern F, T, F, F. The inverse is logically equivalent to the converse, not to the negation.

(B) Converse: "If B, then A"

Same truth table as the inverse: T, T, F, T. Not the negation.

(C) Contrapositive: "If \lnot B, then \lnot A"

Truth table T, F, T, T — identical to the original A \Rightarrow B. The contrapositive is logically equivalent to the original, which is the opposite of being its negation.

Only option (D) A \land \lnot B has the truth table F, T, F, F — exactly the negation pattern.

A walked example

Problem: Write the negation of "If a number is divisible by 6, then it is divisible by 3."

Let A = "divisible by 6", B = "divisible by 3." The statement is A \Rightarrow B.

Negation:

\lnot(A \Rightarrow B) \equiv A \land \lnot B.

In English: "A number is divisible by 6 and is not divisible by 3."

(Such a number does not exist — which is why the original implication is true. The negation is false because no counterexample exists.)

A second example — where the negation is true

Problem: Negate "If it is raining, then the ground is wet."

Let A = "raining", B = "ground wet." The statement is A \Rightarrow B, which is mostly true but has rare counterexamples (a covered patio, a sheltered spot).

Negation:

A \land \lnot B

In English: "It is raining and the ground is not wet." — e.g., standing under a covered patio. If you can exhibit this scenario, the original claim fails.

This is the precise logical form of a counterexample: a single instance of A \land \lnot B is all you need to refute an implication. Not a sweeping denial, not a reverse implication — one specific pairing of A holding and B failing.

Why this matters beyond multiple choice

Proof by contradiction starts here

Proof by contradiction of A \Rightarrow B begins with "assume A is true and B is false, and derive a contradiction." That is, you assume A \land \lnot B. This exact rewrite is the logical foundation of the whole technique. See Proof by Contradiction.

Refuting a universal claim

Every claim of the form "for all x, if P(x) then Q(x)" is refuted by producing a single x for which P(x) \land \lnot Q(x). This is the counterexample pattern in disguise, and it is the negation of an implication at its core.

Reading physics and mathematics textbooks

Theorems of the form "if [condition], then [conclusion]" are broken by finding cases where the condition holds but the conclusion fails. That pattern — the condition holds, the conclusion fails — is A \land \lnot B. The rewrite is not abstract; it is the template for breaking any false theorem.

The exam reflex

Three seconds; one line; full marks.

Related: Logic and Propositions · Converse vs Contrapositive · Proof by Contradiction · Negate a Compound Statement — Invoke De Morgan