Proof techniques
A claim holds in every case you have tried. What does it take to show it holds in every case, and which way of arguing fits which claim?
Where you are 14 inside
What this area is about
A proof is a chain of justified steps from accepted starting points to the exact claim; examples are evidence, not proof. This area covers what a proof may assume and must reach, then the standard methods, one per shape of claim.
How these fit together
What a proof is. Axioms, definitions and theorems says where reasons stop. What may be assumed and what must be shown and writing a proof as prose fix the start and the finish. A conjecture is a claim not yet proved; a counterexample refutes a “for all” claim by one failing case.
Proving “if P, then Q”. Direct proof assumes P and reaches Q. Contraposition proves \neg Q \to \neg P instead. Contradiction assumes the claim false and reaches something impossible. Proof by cases covers every possibility, such as n even or odd. An if and only if is two conditionals, proved separately.
Special tools. Existence proofs show an object exists; the pigeonhole principle is one way to do it. Induction proves P(n) for all integers n \ge n_0 from a base case and a step. Combinatorial proof counts one set two ways.
A path through it
Logic comes first, except the existential quantifier and the quantifier lessons built on it, which follow direct proof. What may be assumed comes before writing a proof as prose. Direct proof comes before contraposition, contradiction, cases, if and only if, induction and pigeonhole, which also needs existence proofs. Counterexamples need quantifiers, existence proofs the existential quantifier, combinatorial proof Pascal’s triangle. Axioms and conjectures need nothing.