IB Maths Proof: Induction, Contradiction & Counterexample Explained

Proof is a Maths AA Higher Level topic (AHL 1.9), tested on both Paper 1 and Paper 2 as structured, multi-mark questions covering induction, contradiction and counterexample. It's rarely a single line — examiners want a base case, an inductive step, or a clear contradiction, each earning its own marks.

Mathematical induction proves a statement true for all positive integers by verifying one base case, then showing that if it holds for n = k, it must also hold for n = k + 1. IB examiners mark each step separately, so skipping the inductive link costs real, checkable marks.

Proof by contradiction assumes the opposite of what you want to prove, then shows that assumption leads to something impossible — like a fraction that turns out not to be in lowest terms. The classic IB example is proving √2 is irrational, and HL students are expected to reproduce it from memory.

Induction proves a statement for every positive integer by building upward from a verified base case. Contradiction instead assumes the statement is false and derives something logically impossible, disproving that assumption. Use induction for claims about all n; use contradiction for irrationality, uniqueness, or impossibility claims.

Formal proof — induction, contradiction and counterexample — sits entirely in syllabus subtopic AHL 1.9, so it's assessed only at Higher Level. Standard Level students still meet the command term 'prove' for direct algebraic or trigonometric identities, but they won't be asked to construct a full inductive or contradiction-based argument.

Proof by exhaustion checks every possible case individually, which only works when the cases are finite. A counterexample disproves a general conjecture with a single failing case — the polynomial n² + n + 41 looks prime for small n but fails at n = 40, since 40² + 40 + 41 = 1681 = 41².

A mark-scheme-safe induction proof has four labelled parts: state and verify the base case, write the inductive assumption for n = k, prove the statement for n = k + 1 using that assumption, then conclude explicitly by the principle of mathematical induction. Missing that final sentence alone can cost a mark.

The mistake I see most is students proving n = k + 1 from scratch instead of using the n = k assumption — that's direct substitution, not induction, and earns almost no marks. The second is skipping the concluding sentence, which the mark scheme often rewards as its own separate point.

Full marks come from precision, not cleverness: write the base case explicitly, state the inductive hypothesis in words, and never jump straight from n = k to the answer without showing the link. Examiners award method marks for correctly using the assumption and accuracy marks for the algebra — skip the link and you lose both.

Induction is examined most often through series (summing expressions like Σr²), divisibility statements (showing an expression divides by a fixed integer), and results involving derivatives or matrices at HL. Contradiction most commonly proves irrationality or the infinitude of primes, and counterexamples usually target a false 'for all n' conjecture.

Proof questions can appear on either Paper 1 or Paper 2 for Maths AA HL, since a proof relies on logical structure rather than calculation, so neither paper favours it specifically. The HL-only Paper 3 investigation can also build an extended, multi-part problem around an inductive or exhaustion-style result.

A full induction or contradiction proof typically carries 5 to 8 marks, split across method and accuracy marks for the base case, the inductive step, and the concluding statement. Losing the concluding sentence alone often costs a single accuracy mark, even when every line of algebra is otherwise correct.

Yes — formal proof by induction, contradiction and counterexample exists only in Maths AA HL; Maths AI doesn't examine these methods at all, focusing instead on applied statistics and modelling. If your child finds abstract, logic-based reasoning genuinely uncomfortable, that's one real reason AI might suit them better than AA HL.

Not entirely, but it helps a lot — proof questions are a reliable source of marks because the four-step structure is learnable, unlike some open-ended modelling questions where the method isn't obvious. A student who reliably nails induction and contradiction can bank 5-8 marks fairly consistently, which matters given grade 7 boundaries often sit in the mid-70s to 80% range.