What a counterexample is: A claim that something holds for all values is destroyed by one value where it fails — that single value is a counterexample. One is enough to disprove a “for all” statement.
One example is enough
Disprove the claim: “every prime number is odd.”
Step by step
- One case is enough. The claim says all primes are odd, so a single even prime would break it.
- Test . Its only factors are 1 and 2, so it is prime — but is even, not odd.
Final answer
is a prime that is not odd, so the claim is false.
Always explain why: It is not enough to name the value — you must explain why it is a counterexample: show it fits the claim's condition and breaks its conclusion.
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Choosing a good counterexample: Hunt where a claim quietly assumes things away: the value , , , negative numbers, fractions between and , and boundary cases. The awkward values are where claims break.
Try the awkward values
Disprove: “for every real number , .”
Step by step
- Whole numbers obey it, so try a fraction. Take .
- Check it breaks the claim. , and , so here.
Final answer
gives , so the claim is false.
Where to look: If whole numbers all obey a claim, the counterexample often hides among fractions or negative numbers.
Negatives break it
Disprove: “if then .”
Step by step
- Try a negative . Take and , so is true.
- Check the conclusion. and , so is false ().
Final answer
, fit but give , so the claim is false.
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An exam-style question
A student claims: “for all real numbers , if then .” Show that this statement is not always true.
Step by step
- Look for the awkward case. The claim feels right for large positive , so test a negative value.
- Check it fits the “if” part. Take : then , which is greater than 9.
- Check it breaks the “then” part. But is not greater than 3.
- Say why this disproves the claim. satisfies yet fails , so the statement is not always true.
Final answer
satisfies but is false, so the statement is not always true.
To get full marks
- Give one clear value, here — a vague “some negative number” is not enough.
- Show it fits the condition: , which is greater than 9.
- Explain it breaks the result: is not greater than 3, so the claim fails for this .
- The value alone, with no check of both parts, can cost marks — the reason is what does the disproving.