Recipe

Disprove a universal claim with a counterexample

Use when: you suspect $\forall x\, P(x)$ is false and need to produce one object that breaks it

Goal: one verified counterexample: an object the claim is actually about, where the claim's conclusion fails

Taught in Counterexamples and disproof

Quick recipe

Identify exactly what the claim requires
Test small or extreme values first
Verify the candidate satisfies the domain, then fails the conclusion
State the counterexample explicitly

1
Break the claim into its domain (what kind of object $x$ ranges over) and its condition (what must hold for that object). A counterexample must fit the domain, then fail the condition.

Example

For "every prime number is odd," the domain is prime numbers, and the condition to fail is "is odd." A counterexample must actually be prime, then turn out even.

Why: misreading the domain or the condition is the single most common way a proposed counterexample turns out invalid.
2
Try the smallest, largest, or otherwise unusual values in the domain before working through it systematically.

Example

Testing primes in increasing order: $2$ , the smallest prime, is worth checking first.

Why: edge cases are disproportionately likely to break a claim that was only checked against typical values.
3
Check both halves separately: confirm the candidate genuinely belongs to the domain the claim is about, then confirm it fails what the claim asserts.

Example

$2$ is prime (fits the domain) and $2$ is even, not odd (fails the conclusion), so $2$ is a genuine counterexample.
- the candidate fits the domain but fails the conclusion — it is a genuine counterexample.
- the candidate fails to fit the domain itself — it says nothing about the claim; discard it and keep searching.
Why: a candidate that never fit the domain in the first place was never a real test of the claim at all.
4
Report the specific object found, not just the fact that one exists. No general argument is needed beyond this one checked case.

Example

$n=2$ disproves "every prime number is odd," since $2$ is prime and $2$ is not odd.

Why: a disproof by counterexample is only as convincing as the one concrete object it names, so the object itself must be stated plainly.

Common mistakes

Proposing an object that fails the domain itself, like using $n=4$ against a claim about primes.
Stopping the search after checking only a few values with no failure, then declaring the claim true.
Giving a general argument for why the claim seems false instead of naming one concrete object.
Testing values in a fixed order without ever trying the small or extreme cases most likely to break the claim.