Logic, proof and discrete math
How does mathematics argue that something is true for every case, and which structures does it argue about first?
Where you are 6 inside
What this area is about
This branch is about how mathematical truth is argued, and about the discrete structures where that arguing is first practiced: the integers, sets, sequences defined by rules, and networks of dots and lines. Here a claim is accepted only when it is proved from definitions and earlier results, never because examples agree.
How these fit together
The tools of argument. Logic covers statements, connectives, quantifiers and valid arguments. Proof techniques turns them into methods: direct proof, contraposition, contradiction, cases, induction and more.
The language. Sets gives the vocabulary of collections, functions and relations, up to the sizes of infinite sets.
Structures to prove things about. Introductory number theory studies divisibility, remainders and common divisors in \mathbb{Z}, through modular arithmetic. Recurrence relations build each term of a sequence from earlier ones and solve some of them in closed form. Introductory graph theory studies vertices and edges: routes, trees, planar drawings and colorings.
A path through it
Logic and proof techniques come first and interleave: most of logic comes before proof, but the existential quantifier, and the quantifier lessons built on it, follow direct proof. Proof techniques come before number theory, sets, recurrences and graph theory. Number theory comes before the equivalence classes in sets, since those use congruence. Sets, which also use De Morgan’s laws from logic, come before graph theory. Recurrences need induction, and can follow proof techniques.
Finish here and you can read and write proofs, the form in which the rest of mathematics is argued.