---
title: "Proof by Contradiction and Counterexample"
description: "Kill a general claim with a single counterexample, and prove one by assuming the opposite and following it until something impossible appears."
canonical: https://lightmysky.com/learn/mathematics/proof-by-contradiction-and-counterexample-mt_pbFTv4v6k8
source: https://lightmysky.com/learn/mathematics/proof-by-contradiction-and-counterexample-mt_pbFTv4v6k8.md
retrieved: 2026-09-12
---

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# Proof by Contradiction and Counterexample

Kill a general claim with a single counterexample, and prove one by assuming the opposite and following it until something impossible appears.

Subject: Mathematics · Area: Mathematical Thinking · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/proof-by-contradiction-and-counterexample-mt_pbFTv4v6k8

## Ready when they can

- Disprove 'every prime number is odd' with one example
- Prove that √2 cannot be written as a fraction
- State clearly what is being assumed at the start of a contradiction proof

## Lesson: One counterexample, one impossible ending

Just like 9 broke every odd number is prime, a for all claim breaks at its first failure. To sink every prime is odd, scan the evens and stop at 2. Check the candidate meets every hypothesis and visibly breaks the conclusion. One case sinks such a claim, but one case never proves one.

**Example.** Assume root 2 is rational in lowest terms, so root 2 = a/b with no common factor. Squaring gives 2b squared = a squared, forcing a even, and writing a = 2k forces b even too. Both even contradicts lowest terms, so the assumption dies and root 2 is irrational.

Contradiction borrows the opposite claim, then throws it. State the assumption up front or nobody can follow the throw. The conclusion always targets the assumption, never the logic: valid steps plus an impossible ending mean a false start.

Long patterns can crack late: n squared + n + 41 gives primes all the way to n = 39, then f(40) = 1600 + 40 + 41 = 1681 = 41 squared. Smallest counterexamples teach most, since they show exactly where the pattern first cracks. One example disproves a for all claim but never proves one.

**Recap.** Sink a general claim with one well checked counterexample, or assume its opposite and follow it to nonsense.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Surds: Simplifying and Rationalising](https://lightmysky.com/learn/mathematics/surds-simplifying-and-rationalising-mt_cqr0crUUXS)
- [Finding a Counterexample](https://lightmysky.com/learn/mathematics/finding-a-counterexample-mt_EoLyw6eP87)
- [Proof by Deduction and Exhaustion](https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f)

## Opens up

- [The Comparison-Sorting Lower Bound](https://lightmysky.com/learn/computing/the-comparison-sorting-lower-bound-mt_7AfUsPV-jW)
- [Asymptotic Notation Made Precise](https://lightmysky.com/learn/computing/asymptotic-notation-made-precise-mt_B0Z6Syy8Ve)
- [The Epsilon-Delta Definition of a Limit](https://lightmysky.com/learn/mathematics/the-epsilon-delta-definition-of-a-limit-mt_jFzRYQzCmk)
- [Statements, Quantifiers and Negation](https://lightmysky.com/learn/mathematics/statements-quantifiers-and-negation-mt_S5NvyPxEZ_)
- [Alternating Series and Absolute Convergence](https://lightmysky.com/learn/mathematics/alternating-series-and-absolute-convergence-mt_sk6F4n1USH)
- [The Pigeonhole Principle](https://lightmysky.com/learn/mathematics/the-pigeonhole-principle-mt_wgfpmZqzd9)