Mathematics
Counterexamples: One Is Enough
One case that fits the conditions but breaks the conclusion kills a statement outright. Why the reverse never works, and what does not count as one.
Counterexample
A single example that satisfies a statement's conditions but not its conclusion, which is enough to prove the statement false.
- Also called
- Counter-example
- Where students meet it
- Grade 8 mathematics, and in the ‘show that this is not always true’ and proof questions on GCSE and IGCSE papers.
The short answer
A counterexample is one case that meets the conditions of a statement but not its conclusion, which is enough to prove the statement false. Against ‘every prime number is odd’, the number 2 settles it in a single line. Collecting examples that fit, however many, proves nothing.
An example
Claim: squaring a number always makes it bigger. Counterexample: 0.5² = 0.25
0.5 is a number, so it meets the condition, and 0.25 is smaller than 0.5, so it fails the conclusion. The claim is now dead. There is no need to try other numbers, and no need to explain why squaring behaves like this — the example alone has done the work.
One kills it; a hundred prove nothing
Disproving and proving are not mirror images. A statement claiming something about every case is destroyed by a single case where it fails, because ‘every’ leaves no room. The same statement can never be established by examples, because there is always another case you have not tried.
The classic warning is the circle. Mark points around a circle, join every pair with a straight chord, and count the regions: 1 point gives 1 region, then 2, 4, 8, 16 — a doubling so convincing that almost everyone predicts 32. Six points give 31. Five examples in a row had agreed, and the pattern was still wrong.
What a question looks like
Exam papers rarely use the word. They ask you to show that a statement is not always true, or say that a student is wrong and ask you to explain why. Both are asking for one clearly written case, not a paragraph of reasoning.
Write down the case, show the arithmetic, and say what goes wrong. For ‘the sum of two prime numbers is always even’, that is: 2 + 3 = 5, and 5 is odd. Three short pieces of information, and the mark is earned.
Say plainly that the statement is false rather than softening it. Writing that it does not always work is weaker than the evidence deserves: the claim covered every case, so one failure has already finished it off.
What does not count
An example that fails the conditions is not a counterexample. Against ‘every multiple of 4 is even’, offering 6 proves nothing, because 6 is not a multiple of 4 and the statement never said anything about it. The case has to be inside the claim before it can break it.
Read the wording closely, because it fences off the territory. ‘Squaring makes a number bigger’ falls to 0.5, but ‘squaring a whole number greater than 1 makes it bigger’ does not — that statement is true, and no counterexample exists.
Common questions
How many counterexamples do I need?
One, written out properly. A second adds no marks and no certainty. What does matter is showing both halves: that your case satisfies the conditions of the statement, and that it fails the conclusion. A case presented without that check often turns out not to be a counterexample at all.
Can a counterexample prove something is true?
No, and nothing can be proved by example. Showing a statement holds in five cases leaves every other case untouched. Proving something true needs an argument covering all cases at once, which usually means algebra — writing an even number as 2n, for instance, so the reasoning applies to every even number.
Does a counterexample have to be a number?
No. Against ‘every rectangle is a square’, a rectangle measuring 3 cm by 5 cm is a perfectly good counterexample, drawn or described. Statements about shapes are disproved by shapes, statements about sequences by sequences. The logic is identical whatever the objects are.
What if I cannot find a counterexample?
Not finding one is not evidence that none exists, but it is a hint the statement may be true. Before giving up, try the awkward cases: 0, 1, negative numbers, fractions between 0 and 1, and the number 2 among the primes. Most school counterexamples are hiding in that short list.
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