Counterexample Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Disprove: 'All prime numbers are odd.'

Solution

  1. 1
    To disprove a universal statement, find a single counterexample.
  2. 2
    Consider 22: it is prime (its only divisors are 1 and 2) and it is even.
  3. 3
    Since 22 is a prime number that is not odd, the statement is false.

Answer

Counterexample: 2 is prime and even.\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.

About Counterexample

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

Learn more about Counterexample →

More Counterexample Examples

Example 2 medium

Disprove: 'For all real numbers [formula], [formula].'

Example 3 easy

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

Example 4 easy

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

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