--- aliases: [counterexample, counterexamples] --- #logic ## Definition > [!tldr] Definition > A **counterexample** is a specific example that shows that a [[Propositions|proposition]] is not always true. Notes: * Another way to view a counterexample, is that it is an example that proves a global, blanket statement about something is false. * A single counterexample is enough to disprove a global statement. * However **no finite number of examples will ever prove a global statement**. ## Examples * The proposition "All mathematicians are male" is false, and a counterexample that proves it would be [any female mathematician you choose](https://exhibits.lib.berkeley.edu/spotlight/women-who-figure/feature/great-women-of-mathematics). - For the proposition "All prime numbers are odd", a counterexample is $p = 2$ because it is a prime number, but it's not odd. This shows that *not all* prime numbers are odd; or in other words the statement "All prime numbers are odd" is false. In this case, $p=2$ is the *only* counterexample. - For the proposition "If $x^2 > 0$ then $x > 0quot;, a counterexample is the number $x = -1$. This shows that the statement is false, because $(-1)^2 > 0$ but $-1 \not > 0$. There are infinitely many other counterexamples -- any negative number will work. ## Resources - Tutorial: [Counterexamples](https://www.khanacademy.org/test-prep/praxis-math/praxis-math-lessons/praxis-math-number-and-quantity/a/gtp--praxis-math--article--counterexamples--lesson) (Khan Academy) - [Another tutorial on counterexamples](https://www.varsitytutors.com/hotmath/hotmath_help/topics/counterexample)