← All Strategies
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Strategy 11 of 12Proof by Contradiction
Assume the opposite — then derive a contradiction
Indirect ProofNegationImpossibility
Assume the opposite of what you want to prove, then show this assumption leads to something impossible (a contradiction). If the negation is false, the original statement must be true. One of the most powerful and widely-used proof techniques.
When to use it
- →When direct proof is difficult
- →When the opposite is easier to analyze
- →When proving existence by ruling out non-existence
- →When the negation gives you extra structure to work with
How to think (step by step)
- 1State what you want to prove
- 2Assume the opposite (negation)
- 3Derive consequences from the assumption
- 4Reach something impossible or contradictory
- 5Conclude: the original statement must be true
Practice Problems
Three problems at increasing difficulty — try each before revealing the hint or solution.
📘Basic
Prove that \(\sqrt{2}\) is irrational.
📙Intermediate
Prove there are infinitely many primes.
📕Advanced
Prove that a real polynomial of odd degree has at least one real root.