Direct proof is the default. You assume p, work forward through definitions and algebra, and arrive at q. This is the technique every proof chapter teaches first, and it is the technique you should always try first.
But sometimes the direct path has no obvious first step. You assume p and stare — "p gives me this, but what do I do with it?" That is the moment to stop pushing and pivot. The pivot is almost mechanical:
Replace p \Rightarrow q with its contrapositive \lnot q \Rightarrow \lnot p. They are logically equivalent. Any proof of the contrapositive counts as a proof of the original.
The reason this rewrite is so useful is that \lnot q — the negation of the conclusion — is often a much more concrete starting point than p. Direct proofs that stall on "assume p" frequently accelerate the moment you restart with "assume \lnot q."
Why the rewrite is legal
The equivalence p \Rightarrow q \equiv \lnot q \Rightarrow \lnot p is a tautology — true in every row of the truth table for p and q. Check it quickly:
| p | q | p \Rightarrow q | \lnot q | \lnot p | \lnot q \Rightarrow \lnot p |
|---|---|---|---|---|---|
| T | T | T | F | F | T |
| T | F | F | T | F | F |
| F | T | T | F | T | T |
| F | F | T | T | T | T |
Columns 3 and 6 match, row by row. So the original and the contrapositive have the same truth value in every scenario — they say the same thing in different words. See Logic and Propositions for the full derivation.
Why the equivalence guarantees your proof counts: if \lnot q \Rightarrow \lnot p is always true, then a proof that it is true in your specific setting automatically shows p \Rightarrow q is true in that setting. The equivalence is the licence — the same licence that backs contradiction and every other proof rewrite.
The reflex — when to pivot
You should consider the contrapositive whenever one or more of these red flags appear in a direct attempt:
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The conclusion is a negation. "Prove that if n^2 + 1 is not a perfect square, then n is not zero." Assume p — "n^2 + 1 is not a perfect square" — and there is no clean definition to unpack. Contrapositive gives "if n = 0, then n^2 + 1 = 1 = 1^2 is a perfect square" — done in one line.
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The hypothesis is a disjunction or negation. "Prove that if n is not even, then n^2 is not even." The direct assumption "n is not even" is less useful than "assume n^2 is even, show n is even" (the contrapositive), which unpacks cleanly with "n must be 2k for some k" — but wait, that is the conclusion. Pivot again to the contrapositive of the contrapositive: "if n is odd, then n^2 is odd." Now "n = 2k+1" gives n^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 — visibly odd. Done.
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The conclusion is "non-existence" or "uniqueness." Direct construction is hard. Negating the conclusion often gives you two concrete objects to work with, from which you derive an impossibility.
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You assumed p and made no progress in 30 seconds. That is the pragmatic red flag. If direct feels stuck, the contrapositive is the first rewrite to try.
A three-line worked example
Claim. If n^2 is even, then n is even.
Direct attempt. Assume n^2 is even. Write n^2 = 2m for some integer m. You want to show n = 2k for some k. Can you do that from n^2 = 2m? Not without extra work — the square root of 2m is not obviously even. The direct path stalls.
Pivot to contrapositive. Prove instead: if n is odd, then n^2 is odd.
Assume n is odd: n = 2k + 1 for some integer k.
Compute: n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1. This is odd.
Conclude. By the contrapositive equivalence, the original claim — "if n^2 is even, then n is even" — is proved. \blacksquare
Three lines after the pivot. The direct attempt had no similar three-line version.
An interactive: when to choose which
Drag the dot across four proof scenarios. For each, the readout indicates whether direct or contrapositive is the cleaner first approach.
Why contrapositive is not the same as contradiction
Students routinely conflate the two. They are cousins, not twins.
- Contrapositive proves \lnot q \Rightarrow \lnot p directly. You assume \lnot q, walk forward, arrive at \lnot p. The conclusion is \lnot p itself.
- Contradiction proves p \Rightarrow q by assuming p \land \lnot q and deriving any contradiction. The conclusion is an absurdity like 1 = 0 or n is both even and odd.
Contrapositive is tidier and should be tried first when the direct approach stalls. Fall back to contradiction only if the contrapositive itself does not land cleanly. See Proof by Contrapositive for the full technique.
The exam reflex
- Try direct. Assume p, look for the first definition to unpack.
- If you are stuck in 30 seconds, check the form of p and q:
- Is the conclusion a negation? Pivot to contrapositive.
- Is the hypothesis a negation? Pivot to contrapositive.
- Is the conclusion a disjunction? Consider contrapositive — its negation is a conjunction, which gives you two usable premises.
- Write: "By the contrapositive, it suffices to prove: if \lnot q, then \lnot p."
- Continue as a direct proof of the new statement.
- Conclude: "By contrapositive equivalence, the original statement holds. \blacksquare"
That is the whole move. One rewrite, one equivalence, and a stalled proof becomes a three-line win.
Related: Logic and Propositions · Proof by Contrapositive · Converse vs Contrapositive — Why They Are Not the Same Thing · Proof by Contradiction