--- title: "Counterexamples and Testing a Conjecture" description: "Before proving a statement, spend real effort trying to break it. A counterexample settles the question outright, and a failed attempt usually shows which hypothesis the proof will need." canonical: https://lightmysky.com/learn/mathematics/counterexamples-and-testing-a-conjecture-mt_SjA4jqNkaM source: https://lightmysky.com/learn/mathematics/counterexamples-and-testing-a-conjecture-mt_SjA4jqNkaM.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Counterexamples and Testing a Conjecture Before proving a statement, spend real effort trying to break it. A counterexample settles the question outright, and a failed attempt usually shows which hypothesis the proof will need. Subject: Mathematics · Area: Mathematical Thinking · Ages 22 to 24 Page: https://lightmysky.com/learn/mathematics/counterexamples-and-testing-a-conjecture-mt_SjA4jqNkaM ## Ready when they can - Construct a counterexample to a plausible but false statement - Turn a failed counterexample into the hypothesis the theorem was missing - Test a conjecture on the smallest cases and the most awkward ones first ## Lesson: Breaking claims to make them true Many conjectures claim something about all members of a set: all primes, all continuous functions, all groups with some property. One counterexample defeats such a claim: a single object meeting the hypotheses but breaking the conclusion. The denial of all A are B is some A is not B, so build exactly that object. For if P then Q, hunt the single falsifying row: true hypothesis with false conclusion. **Example.** Claim: all prime numbers are odd. The number 2 is prime and even, so it meets the setup but breaks the result, and the claim falls. Note that 9, 15, and 1 cannot break it, since none of them is prime at all. Claim: the product of any two irrationals is irrational. Root 2 times root 2 equals 2, which is rational, so that pair breaks it while root 2 times root 3 stays irrational. A failed attempt is still useful: it usually shows which hypothesis the proof will need. Trying hard to break the claim that differentiability forces continuity always collapses at the same place, since every candidate turns out continuous. That pattern signals the hypothesis is doing the real work. Likewise, pairs that cancel like g equals minus f show why the sum of two discontinuous functions can be continuous, and the repaired claim is: a continuous plus a discontinuous function is discontinuous there. **Tip.** Test conjectures on the smallest cases and the most awkward ones first: zero, one, empty, discontinuous, extreme. For n squared plus n plus 41, try 0 giving 41, then 1 giving 43, then the awkward 40 giving 1681, which is 41 times 41. Large random cases usually behave well and rarely expose the flaw, so cheap small tests find the most breaks. **Recap.** One good counterexample settles a universal claim, and near misses reveal the missing hypothesis. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Definitions as Design Decisions](https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d) - [Continuity and Uniform Continuity](https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz) ## Opens up - [Open Problems as a Genre](https://lightmysky.com/learn/mathematics/open-problems-as-a-genre-mt_qBWcaUWG9J)