The main types of proof

Proving a mathematical proposition is not about convincing yourself intuitively that it is true: it means producing a rigorous chain of implications, starting from accepted or already proven hypotheses, all the way to the desired conclusion. There are several classic strategies, and knowing which one to choose is already a skill in its own right.

Direct proof

It starts from the hypotheses and chains deductions up to the conclusion: if $P$ is true, we show step by step that $Q$ necessarily follows. It is the most natural method, but not always the easiest to carry out.

Proof by contrapositive

It exploits a fundamental logical equivalence: $P \implies Q$ is strictly equivalent to its contrapositive $\neg Q \implies \neg P$.

When it is hard to start directly from $P$, it is often easier to assume $Q$ is false and show that $P$ must then be false too.

Be careful never to confuse the contrapositive, which is always equivalent to the original implication, with the converse $Q \implies P$, which in general has no guaranteed truth relationship with it.

Proof by contradiction

We assume the opposite of what we want to prove, then deduce a contradiction from it — a proposition and its negation both true at the same time. Since this contradiction is impossible, the initial assumption was false, and the original proposition is therefore true.

Step	What we write
$1$	we want to prove $P$
$2$	we assume $\neg P$
$3$	we carry out valid logical deductions
$4$	we obtain $Q \wedge \neg Q$, a contradiction
$5$	so $\neg P$ is impossible
$6$	conclusion: $P$ is true

It is the method of choice for proving that an object does not exist, or that a given number is irrational.

Proof by cases

It splits the problem into several situations that together cover all possibilities — for example an integer being even or odd — and proves the proposition separately in each case.

Two points must be checked with the greatest care: that the cases cover all possibilities, without exception, and that each one is handled correctly all the way through.