What Is a Counterexample?

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

A counterexample is a specific example that shows a universal statement is false.

A universal statement looks like:

For all x in some set, P(x) is true.

To disprove it, you only need to find one x such that P(x) is false.

Example:

The statement "All prime numbers are odd" is false because 2 is a prime number that is not odd.
The statement "If a·b = 0, then a = 0" is false because 2·0 = 0 even though 2 ≠ 0.

Counterexamples are powerful because they give insight into why a statement fails.

TL;DR — key idea

A counterexample is a single specific instance that makes a universal statement false.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
Explain in your own words why a single counterexample is enough to disprove a universal statement. Then provide your own counterexample to the incorrect statement: "If a·b = 0, then both a and b must be 0."

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.