Logical Reasoning · 3.1.6
Counter-examples to statements
This standard asks students to find one specific case that makes a given general statement false. Finding just a single valid counter-example is enough to prove the statement false overall, even though the statement may hold true in many other cases that were not chosen.
The official learning standard (3.1.6)
“Determine a counter-example to negate the truth of a particular statement.”
What it means
This standard asks students to find one specific case that makes a given general statement false. Finding just a single valid counter-example is enough to prove the statement false overall, even though the statement may hold true in many other cases that were not chosen.
How it is examined
Paper 1 often gives a general (usually false) statement, such as one about all prime numbers or all multiples of a number, and asks students to state a counter-example. Paper 2 may require students to use a counter-example to justify why a given statement or argument is not always true.
Worked example
Determine a counter-example to show that the statement 'All prime numbers are odd numbers' is false.
- The statement claims every prime number is odd
- Consider the number 2
- 2 is a prime number, since its only factors are 1 and 2
- 2 is an even number, not odd
- Since 2 is prime but not odd, it contradicts the statement
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How many counter-examples do I need to disprove a statement?
Just one. A single valid case where the statement fails is enough to prove the general statement is false, no matter how many other cases make it true.
Can I use a counter-example to prove a statement is true?
No. A counter-example only disproves a statement by showing an exception.
To prove a general statement true for all cases, you need a general proof or reasoning, not just checking examples.
What should I check first when looking for a counter-example?
Try unusual or extreme cases first, such as zero, one, negative numbers, or the smallest or largest value in the set being discussed, since general statements often fail at these boundary cases.