To test a claim about numbers, first look for a counterexample. One case where the claim fails disproves it, and no number of cases where it works proves it.
This page belongs to the logical reasoning chapter. It applies the skill from testing deductive arguments with counterexamples to a claim you might hear outside the classroom.
Which way should I test a claim?
Read the claim and ask whether it says “always” or “sometimes”. A claim that says “always” can be broken by one exception, so look for one. A claim that says “sometimes” is proved by one example, but disproving it means showing that every case fails.
- Write the claim as a clear statement with “all” or “some”.
- Try the small and awkward cases: 0, 1, 2, a negative number.
- If one fails, stop and write it as a counterexample.
- If none fail, look for a general reason that covers every case.
Worked example: does squaring always make a number bigger?
A friend says, “The square of a number is always bigger than the number itself.” Try it.
Test 3: 3² = 9, which is bigger than 3. Test 5: 25 is bigger than 5. Test 10: 100 is bigger than 10. Three checks agree with the claim.
Now try the small cases. Test 1: 1² = 1, which is equal to 1, not bigger. The claim fails, so one counterexample already disproves it.
Test 0.5 as well: 0.5² = 0.25, which is smaller than 0.5. A second counterexample confirms the fault, although one was enough.
The corrected claim is “For any number greater than 1, the square is bigger than the number.” To see why it holds, note that n > 1 and multiplying both sides by n gives n² > n.
What goes wrong when students test claims?
A common slip is to stop after a few easy cases and call the claim proved. The three examples above all agreed, yet the claim was false.
| Step | Wrong | Right |
|---|---|---|
| Cases tried | 3, 5, 10 | 3, 5, 10, then 1 and 0.5 |
| Conclusion | Claim is true | Claim is false, 1 is a counterexample |
| What the examples show | proof | a pattern that needs a general reason |
The habit that fixes this is simple: always include 0, 1 and a fraction in the cases you try.
Where does this skill appear in the chapter?
The same idea drives negating statements with quantifiers, because the negation of “all” is “some … not”, and a counterexample is exactly that “some”. It also supports questions on the strength of inductive arguments.
Work the logical reasoning practice set to see the pattern in exam wording. For lessons with a teacher, see online one-to-one Mathematics tuition, and use the algebra step repair trainer if the algebra in a general argument is where you slip.