Logic & Set Theory
The grammar of mathematics. Before you can prove anything, you need to know what a proposition is, how to chain claims into arguments, and what kind of object a set is. This chapter is the toolkit every other chapter quietly relies on.
Propositional Logic
Propositions, the five connectives, truth tables, tautologies and equivalences — the calculus of true and false.
Sets & Set Operations
Cantor's naive view, set notation, union, intersection, complement, Cartesian products — and Russell's paradox as a teaser for why axioms matter.
Proof Techniques
Direct, contrapositive, contradiction, cases, and induction. Each with the logical structure spelled out and a classic worked example.
Relations & Functions
The set-theoretic view of relations and functions. Equivalence relations, partial orders, injections, surjections, bijections, composition.
Voting Theory
Plurality, Borda, Condorcet, instant-runoff, approval — and Arrow's theorem on why no method can be fully fair.
Apportionment Methods
Hamilton, Jefferson, Webster, Adams, Huntington-Hill — dividing whole seats among unequal groups, and the paradoxes you can't dodge.