Proofs

In mathematics it is not enough that a rule works for a few examples. A statement only holds once it is certain for all cases. A proof is the gapless justification for that. We look at three tools: two for proving a statement and one for refuting it.

What a proof has to do

A proof leads from things already accepted as true, in clear steps, to the new claim. Every step must be compelling so that no doubt remains at the end. Examples can suggest a conjecture, but they never prove that something holds for infinitely many cases.

The direct proof

In a direct proof you start from the assumptions and calculate straight ahead to the claim. A classic: the sum of two even numbers is even again. Every even number is twice a whole number: one of them can be written as $2m$, the other as $2n$, with whole numbers $m$ and $n$. Adding both and factoring out the 2 shows at once that the result is again twice a whole number. The proof therefore covers all even numbers at once, not just the ones you happened to try.

The counterexample

To show that a claim is false, a single case where it fails is enough. This case is called a counterexample. If someone claims all prime numbers are odd, the 2 suffices: it is a prime number and yet even. That settles the claim, no matter how many other primes are odd.

The proof by contradiction

In a proof by contradiction you assume the claim is false and follow this assumption until something impossible comes out. Since the steps were correct, the false assumption itself must be the mistake. So the claim was true after all. This method feels unusual at first, but it is very powerful.

A simple example: there is no largest even number. Assume the opposite, that there is one, and call it $g$. Then $g + 2$ is even as well and larger than $g$. So $g$ was not the largest after all, and the assumption collapses. There really is no largest even number.