To disprove an “all” claim you need one counterexample, a single case where the claim fails. You do not need to check every number.
This lesson is part of proving or disproving an everyday mathematical claim. The chapter background is in logical reasoning.
Where do I look for a counterexample?
Go through a short search order and stop at the first case that fails.
- Try 0 and 1, which behave differently from other numbers.
- Try 2, the only even prime, and other small values.
- Try a negative number and a fraction.
- Keep going through the next whole numbers, because a pattern can break late.
Write each trial in a line, so the failing one is easy to point to.
Worked example: a claim that looks safe
A classmate says, “For every whole number n, the value n² + n + 11 is a prime number.” Test the search order.
Try n = 0: 11, which is prime. Try n = 1: 13, prime. Try n = 2: 17, prime.
Then n = 3 gives 23, n = 4 gives 31, n = 5 gives 41, n = 6 gives 53, n = 7 gives 67, n = 8 gives 83 and n = 9 gives 101. All are prime.
Now try n = 10: 100 + 10 + 11 = 121. Since 121 = 11 × 11, it is not prime.
The claim fails at n = 10. Ten cases looked fine, and one was enough to end the argument.
The mistake that costs marks
The common slip is stopping at the first few successes and writing “the claim is true”. It feels safe because the arithmetic is correct each time.
| Step | Wrong | Right |
|---|---|---|
| Cases tried | n = 0, 1, 2 | n = 0 to 10 |
| All primes so far | claim is true | keep going, cases do not prove a claim |
| Case n = 10 | not reached | 121 = 11 × 11, so not prime |
| Conclusion | claim is true | claim is false, n = 10 is a counterexample |
A short list of successes never proves an “all” claim, however tidy it looks.
Check yourself
A student claims, “For every number x, (x + 2)² = x² + 4.” Find a counterexample.
Answer
Try x = 1. The left side is (1 + 2)² = 9. The right side is 1² + 4 = 5. Since 9 is not equal to 5, the claim fails.
The correct expansion is (x + 2)² = x² + 4x + 4. The missing 4x is the term the student forgot, and at x = 0 it would not show, which is why 0 is a poor test value here.
What to study next
Move on to separating a valid argument from a true conclusion reached incorrectly. When you are ready, try the integrated practice set.
The algebra step repair trainer helps if expansion slips like the one above are common for you. For a teacher to work through claims with you, see online one-to-one Mathematics tuition.