Direct Proof and Disproof (AA SL)
A direct proof turns a general statement about numbers into an algebraic expression, then manipulates that expression until the required conclusion is reached, with every step justified. A disproof works the opposite way round - one well-chosen counterexample is enough to break a false claim. This page covers how to set both out cleanly, with worked examples and the mistakes that lose the most marks. It's part of the broader Proof topic.
35 questions on this sub-topic.
Setting a proof out correctly
Covered under IB syllabus reference SL1.6: simple deductive proof, numerical and algebraic, laid out from a left-hand side (LHS) to a right-hand side (RHS), with correct use of the equality and identity symbols. There's no formula to memorise here - the skill is in the algebraic set-up and the logic that closes the proof.
General integer forms
Even: \(2n\) Odd: \(2n+1\) Consecutive: \(n, n+1\)
Every direct proof about integers starts by writing the given condition (even, odd, consecutive, a multiple of some number) as an algebraic expression with \(n \in \mathbb{Z}\).
LHS to RHS layout
Start → manipulate → state the conclusion
Work from the given side of the statement towards the side you're trying to reach, showing each algebraic step. Finish with a sentence that explicitly states why the result follows - the algebra alone is not the proof.
Need the full syllabus wording and worked proof techniques for the wider topic? See Proof.
Worked examples
Prove that the sum of any two consecutive integers is odd.
Worked solution
Let the two consecutive integers be \(n\) and \(n+1\), where \(n \in \mathbb{Z}\). Using \(n\) and \(n+1\) captures “consecutive” in a single variable. M1
\(n + (n+1) = 2n + 1.\) A1
\(2n\) is even (a multiple of 2), so \(2n+1\) is one more than an even number and is therefore odd for every integer \(n\). R1
Prove that \(a^2 + b^2 \ge 2ab\) for all real numbers \(a, b\).
Worked solution
For any real \(a,b\), a square is never negative: \((a-b)^2 \ge 0.\) M1
\(a^2 - 2ab + b^2 \ge 0.\) A1
Adding \(2ab\) to both sides: \(a^2 + b^2 \ge 2ab.\) A1
The step is reversible and valid for all real \(a,b\), with equality exactly when \(a=b\). R1
Common mistakes
- Forgetting the reasoning line at the end. Reaching \(4(2n+1)\) isn't itself a proof of divisibility by 4 - you also need to state that \(2n+1\) is an integer, so the whole expression is a multiple of 4.
- Trying to "prove" a false statement instead of disproving it. If a question asks you to test or disprove a claim, look for a counterexample first - don't waste time attempting a general proof of something that isn't actually true for every case.
- Assuming the result to prove it. Starting from the equation you're trying to establish and working backwards to something true is circular unless every step is shown to be reversible - it's safer to start from the given facts and work forwards to the conclusion.
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34 direct proof and disproof questions, marked instantly like the real exam.
Quick answers
What is a direct proof in IB Maths?
A direct proof starts from a general algebraic statement of the given facts (for example, letting an even integer be \(2n\)) and manipulates it step by step until the required result is reached, with each step justified.
How do you disprove a statement in IB Maths?
To disprove a statement you only need one counterexample - a single case where the statement fails. You do not need to show it fails in general, just find one value that breaks it.