---
tag: 0ED72CTBNP
kind: concept
name: Counterexample
msc: [00A35]
uses: [0B88PQTJ66, 0P3A06QNDP]
sources:
  - url: https://en.wikipedia.org/wiki/Counterexample
    locator: lead section
same_as:
  wikidata: Q596077
  wikipedia: Counterexample
---

## Summary

A counterexample to a universal statement $\forall x \in X, \, P(x)$ is an element $a \in X$ such that $P(a)$ is false.
One counterexample shows that the universal statement is false.

## Remarks

The number $2$ is a counterexample to "each prime number is odd".
This works because of the [[negation of a universal statement|0P3A06QNDP]]: $\neg \forall x \, P(x)$ is equivalent to $\exists x \, \neg P(x)$.

## Explanations

### L1 human

To show that a general claim is wrong, one bad case is enough.
"Each continuous function is differentiable" is false: $f(x) = |x|$ is continuous but not differentiable at $0$.