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A-Level Mathematics • Proof • A4

Disproof by Counterexample

Learn how one valid counterexample is enough to disprove a universal mathematical statement.

Video Lesson 11 Minutes Pure Mathematics Advanced Level

🎬 Animated Lesson

🎙 Narration

Watch how a single carefully chosen example can show that a statement claimed for every case is false.

🧠 Key Idea

To disprove a universal statement, it is enough to find one case in which the statement fails.

✅ Learning Outcome

After completing this lesson you should be able to:

  • Recognise universal statements
  • Search for an appropriate counterexample
  • Verify that the counterexample satisfies the conditions
  • Explain clearly why the statement is false

Worked Example

Disprove: All prime numbers are odd.

The number 2 is a prime number because its only positive factors are 1 and 2.

However, 2 is even, not odd.

Therefore, 2 is a counterexample.

Hence, the statement “all prime numbers are odd” is false.

Quick Quiz

How many counterexamples are needed to disprove a universal statement?

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