What does counter example mean in math terms?

Answer 1

It is an example that demonstrates, by its very existence, that an assertion is false. Usually experience suggests that the assertion is true: there is a large amount of supporting "evidence" but the statement has not been proven. The counter-example, though demolishes the assertion

For example:

Assertion: all prime numbers are odd.

Counter example: 2. It is a prime but it is not odd.

Therefore the assertion is false.

This was a favourite "trap" at GCSE exams in the UK.

Assertion: if you divide a nuber it becomes smaller.

Counter example 1: 2 divided by a half is, in fact, 4.

Counter example 2: -10 divided by 2 is -5 (which is larger by being less negative).