← All Strategies
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Strategy 03 of 12Counterexamples
One example says YES — one counterexample says NO
DisproveFalse ClaimsSearch
A counterexample is the most direct method to disprove a claim. Instead of explaining why something is wrong, simply find one instance where it fails. One example is enough to refute a universal claim.
When to use it
- →When the claim says 'for all' or 'always'
- →When you suspect a claim is false
- →When asked to investigate whether something holds
- →When you want to quickly test a hypothesis
How to think (step by step)
- 1Understand the claim precisely: What exactly does it assert?
- 2Look for suspicious values: Which values might break it?
- 3Try special cases: Small, large, negative, zero...
- 4Verify carefully: Once found, double-check the counterexample
Practice Problems
Three problems at increasing difficulty — try each before revealing the hint or solution.
📘Basic
A student claims: 'If \(a^2 = b^2\), then \(a = b\) for every real \(a, b\).' Is this true? If not, give a counterexample.
📙Intermediate
Is it true that for all integers \(n \geq 2\), the number \(2^n - 1\) is prime?
📕Advanced
Is it true that every continuous function \(f: [0,1] \to \mathbb{R}\) that is differentiable on \((0,1)\) with \(f'(x) > 0\) must be strictly increasing on \([0,1]\)?