1.15 1.15 Proof by induction, contradiction and…

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Topic 1.15 · 1. Number and Algebra

01Work them in this order

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Study Notes

Quick answer

1.15 Proof by induction, contradiction and counterexample in IB Mathematics: Analysis and Approaches HL includes 5 focused study resources covering the core ideas, revision points, and study support you need to review this topic efficiently.

What this topic means

Proof by induction, contradiction and counterexample in IB Math AA HL focuses on constructing logically complete mathematical arguments using accepted proof structures. It belongs to Number and Algebra, where success depends on selecting efficient algebraic methods and communicating exact or appropriately rounded results. The Analysis and Approaches course emphasises algebraic reasoning, exact methods, and justified mathematical steps. At HL, be prepared to connect the idea with earlier syllabus content and sustain a multi-step argument.

Questions on Proof by induction, contradiction and counterexample reward a precise statement, declared assumptions, a logically sufficient chain of justified steps, and a conclusion that matches what had to be proved. Numerical examples can test an idea but do not replace a general proof.

02Common questions

What is 1.15 Proof by induction, contradiction and counterexample in IB Mathematics: Analysis and Approaches HL?

Proof by induction, contradiction and counterexample in IB Math AA HL focuses on constructing logically complete mathematical arguments using accepted proof structures. It belongs to Number and Algebra, where success depends on selecting efficient algebraic methods and communicating exact or appropriately rounded results. The Analysis and Approaches course emphasises algebraic reasoning, exact methods, and justified mathematical steps. At HL, be prepared to connect the idea with earlier syllabus content and sustain a multi-step argument. 1.15 Proof by induction, contradiction and counterexample sits within 1. Number and Algebra in the IB Mathematics: Analysis and Approaches HL syllabus structure.

How do I study 1.15 Proof by induction, contradiction and counterexample for the IB exam?

Learn the definitions and proof structures, annotate why each line follows, reconstruct complete arguments without notes, and practise identifying hidden assumptions, missing cases, and valid counterexamples.

← Previous1.14 De Moivre's theorem, powers and roots of complex numbers Next →1.16 Systems of linear equations

1.11 Partial fractions

1.12 Complex numbers in Cartesian form and the Argand diagram

1.13 Modulus–argument (polar) and Euler form

1.14 De Moivre's theorem, powers and roots of complex numbers

1.16 Systems of linear equations

Topic essentials

State the claim and assumptions precisely, justify each implication, and finish with the required conclusion.
Connect Proof by induction, contradiction and counterexample to the wider Number and Algebra unit and explain why the selected method is valid.
Look for a missing case, an unproved converse, or an example that could disprove a universal statement.

Key concepts to master

State the proposition, assumptions, variable domain, and the conclusion required for Proof by induction, contradiction and counterexample.
Choose a direct, contrapositive, contradiction, induction, or counterexample structure only when it fits the claim, and justify every transition.
Check that the reasoning covers all permitted cases and that no step assumes the result it is trying to prove.

How to study this topic

1
Build a Proof by induction, contradiction and counterexample checklist of definitions, allowed assumptions, proof structures, and logical statements such as implication, equivalence, necessity, and sufficiency.
2
Annotate one complete proof line by line, naming the fact or earlier result that justifies each step, then reconstruct it without looking.
3
Practise a related claim that needs a different proof structure and use a counterexample search to test whether the statement is actually true.

Common mistakes

Checking several examples and presenting the pattern as proof of a statement intended to hold generally.
Skipping an implication, using the desired conclusion as an assumption, or citing a theorem without showing that its conditions hold.
Proving only one direction of an “if and only if” statement or failing to cover a base case, boundary case, or permitted variable domain.

Study prompts

Write a complete proof for a claim involving Proof by induction, contradiction and counterexample, stating the assumptions and justifying every implication.
Compare two possible proof structures for a Proof by induction, contradiction and counterexample claim and explain which is more direct or reliable.
Find the first invalid or unsupported step in a proposed proof and repair the argument.
Decide whether a related statement is always true; prove it or give a valid counterexample.