Counterexample Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Counterexample.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A counterexample is a specific instance that satisfies the hypothesis of a claim but contradicts its conclusion, thereby disproving the general statement.

One case where it fails is enough to kill a 'for all' claim.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: To disprove \forall x\, P(x), find one x where P(x) is false.

Common stuck point: A counterexample disproves "for all" claims, but finding many examples that work does NOT prove a universal statement is true.

Sense of Study hint: Try small, simple values first (0, 1, 2, -1, 1/2). Counterexamples are usually lurking among the simplest cases.

Worked Examples

Example 1

easy
Disprove: 'All prime numbers are odd.'

Solution

  1. 1
    To disprove a universal statement, find a single counterexample.
  2. 2
    Consider 2: it is prime (its only divisors are 1 and 2) and it is even.
  3. 3
    Since 2 is a prime number that is not odd, the statement is false.

Answer

\text{Counterexample: } 2 \text{ is prime and even.}
A counterexample is a specific case that shows a universal statement is false. Only one counterexample is needed to disprove a 'for all' claim.

Example 2

medium
Disprove: 'For all real numbers x, x^2 > x.'

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find a counterexample to disprove: 'The sum of any two prime numbers is even.'

Example 2

easy
Find a counterexample to disprove: 'If an integer is divisible by 6, then it is divisible by 4.'
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Background Knowledge

These ideas may be useful before you work through the harder examples.

quantifiers