Counterexamples and disproof

Objective. Disprove universal claims with decisive examples.

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Learn: the key idea

A counterexample is a single specific case that disproves a general claim, since one failure is enough to show a 'for all' statement is false. To disprove a conjecture, search for values (often small integers, negatives, zero or fractions) that break the stated pattern, then clearly state why that example fails the claim. For example, the claim 'n² is always greater than n' fails for n = 1 (1² = 1, not greater) and for n = 0.5 (0.25 < 0.5), so a single such value disproves the universal statement. Testing edge cases like negative numbers or zero is often the fastest way to find a counterexample.

Worked example

Give a counterexample to disprove the claim that the sum of two prime numbers is always even.

1 $Recall the claim: sum of two primes is always even$
2 $Choose primes 2 and 3$
3 $Compute the sum: 2 + 3 = 5$
4 $Since 5 is odd, not even, this disproves the claim$

Answer: 2 + 3 = 5 (odd), disproving the claim

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Common mistake. Students sometimes try to disprove a claim by giving a vague explanation instead of a concrete counterexample, e.g. saying 'it isn't always true' without stating a specific value; a disproof must show one exact case that fails.

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