Direct Proof and Disproof (AA HL)
Not every proof needs the machinery of induction. Plenty of IB statements are settled directly - by writing the general case in symbols and reasoning forward - or by contradiction, or knocked down entirely with a single well-chosen counterexample. This page covers those three everyday tools, with worked examples and the logical slips that undo an otherwise correct argument. It's part of the broader Proof topic.
39 questions on this sub-topic.
Three proof structures
Covered under IB syllabus reference AHL1.15, alongside proof by induction. None of these is a formula in the booklet sense - they're patterns of argument, and picking the right one for a given statement is half the skill.
Direct proof
Write the general case → manipulate algebraically → reach the stated result
Not in the booklet - a method. Start from arbitrary integers or reals (never a specific number), and work forward until you arrive at exactly what was asked for.
Proof by contradiction
Assume the negation → derive a contradiction → conclude
Not in the booklet. Assume the opposite of what you want to prove is true, then show that assumption forces something impossible.
Disproof by counterexample
State the failing case → explain why it fails
Not in the booklet. One specific, correctly-checked case is enough to disprove a "for all" claim - no general argument needed.
Statements about all positive integers - not just a starting few - are often better suited to mathematical induction instead; see the Proof page for GDC pointers on checking your working numerically first.
Worked examples
Prove that the sum of any two even integers is even.
Worked solution
Let the even integers be \(2a\) and \(2b\) for \(a, b \in \mathbb{Z}.\) M1
Their sum is \(2a + 2b = 2(a+b)\), A1
a multiple of 2, hence even. R1 AG
Prove that for an integer \(n\), if \(n^2\) is even then \(n\) is even. (Hint: prove the contrapositive.)
Worked solution
if \(n\) is odd then \(n^2\) is odd. M1
if \(n\) is odd, \(n = 2k+1\), A1
so \(n^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1\) M1
which is odd. A1
Since the contrapositive is true, the original statement holds. R1
Common mistakes
- "Proving" a general statement with one example. Checking that a claim works for \(n=4\) shows nothing about \(n=5\) or any other value - a direct proof needs general variables (like \(2a\), \(2b\), or \(2k+1\)), not a single number.
- Asserting a counterexample instead of checking it. Naming a value that you believe breaks a statement isn't enough - the disproof mark comes from actually substituting it in and showing, in full, that it fails.
- Forming the contrapositive incorrectly. The contrapositive of "if \(P\) then \(Q\)" is "if not \(Q\) then not \(P\)" - negating only one part, or reversing the implication without negating (which gives the converse instead), proves nothing about the original statement.
Ready to practise properly?
41 direct-proof and disproof questions, marked instantly like the real exam.
Quick answers
What is the difference between a direct proof and a proof by contradiction?
A direct proof starts from general assumptions and reasons forward, step by logical step, to the stated result. A proof by contradiction instead assumes the result is false, then shows that assumption leads to something impossible - which means the original result must be true after all.
How do you disprove a statement in IB Maths?
A general statement (one claimed to be true for all cases) is disproved by finding a single counterexample - one specific case where it fails - and clearly checking that the case really does break the statement.