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MathidiaMathematics Academy
Mathematical research methods
Lesson 69 of 75
Challenge

Counterexamples

Objective. Stress-test assumptions and repair false claims.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
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Learn: the key idea

Counterexamples disprove a false general claim by exhibiting one specific case where the claim fails, and are the standard way to stress-test a conjecture or repair an overly broad statement. For example, the claim "every continuous function is differentiable" is false, with f(x) = |x| as a counterexample since it is continuous everywhere but not differentiable at x = 0. Finding a counterexample often means testing edge cases: zero, negative numbers, or boundary points of a domain, since false claims frequently fail precisely at these special cases. Once found, a counterexample also suggests how to repair the claim, for instance by adding the missing hypothesis that excludes the counterexample.

Worked example

Find a counterexample to the claim: "for all real numbers x, if x² > 0 then x > 0".

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  1. 1Identify the claim's structure: it asserts x² > 0 implies x > 0 for every real x.
  2. 2Test a negative value, such as x = −2.
  3. 3Compute x² = (−2)² = 4, which is greater than 0.
  4. 4Since x = −2 satisfies x² > 0 but not x > 0, x = −2 is a valid counterexample.

Answer: x = −2

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Common mistake. A common mistake is testing only typical, well-behaved cases when searching for a counterexample and concluding a false claim is true because those cases happen to satisfy it; edge cases and boundary points must also be checked.

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