Proof and Logic

Patterns that hold for forty cases, or nine hundred million, and then fail, what a proof is instead, and a first proof from the definitions of even and odd.

Sentences with a truth value, the five connectives that combine them, why a conditional with a false hypothesis counts as true, and the converse and contrapositive told apart.

"For all" and "there exists", why the domain is part of every claim, how to negate a quantified statement, and why swapping two quantifiers can turn a true statement into a false one.

How to prove "if $P$ then $Q$" and "for every $x$" from definitions alone, with divisibility and parity as working material, proofs that split into cases, and proofs that something exists with or without naming it.

Two indirect methods for statements the direct route cannot reach, proving that if $n^2$ is even then $n$ is even, that $\sqrt{2}$ is irrational and that the primes never run out, and how to tell which method a proof really needs.

Proving a statement for every natural number from two finite checks, why the well-ordering of $\mathbb{N}$ justifies it, sums, inequalities and divisibility, strong induction for prime factorisation and Fibonacci numbers, and the proof that all horses are one colour, taken apart.

Membership, set builder notation, subsets, union, intersection, difference and complement, proving two sets equal by double inclusion, De Morgan's laws, power sets and products, and the paradox that forced set builder notation to name a surrounding set.

A function as a rule with a stated domain and codomain, image and preimage, injective, surjective and bijective maps with a proof of each, composition, the theorem that a function has an inverse exactly when it is a bijection, and how many functions join two finite sets.

Relations as sets of pairs, the reflexive, symmetric and transitive properties, the theorem that equivalence classes partition a set, congruence modulo $n$ and the fractions built as classes, and why an operation on classes must be shown well defined.

Antisymmetry and partial orders, with $\le$, $\subseteq$ and divisibility proved to be orders, total and strict orders, Hasse diagrams, maximal against greatest elements, upper bounds, and the least upper bound that the rationals sometimes lack.

Size defined by bijections, Hilbert's hotel, the countability of the integers, the pairs of natural numbers and the rationals, Cantor's diagonal argument that the reals are uncountable, and Cantor's theorem that every set is smaller than its power set.

The completeness axiom as the one property that separates the reals from the rationals, and the Archimedean property, the density of the rationals and the existence of $\sqrt{2}$ proved from it.

Convergence of a sequence written with $\epsilon$ and $N$ and proved from the definition, from $\frac{1}{n}\to 0$ to uniqueness, sums, boundedness and the monotone convergence theorem, which holds in the reals and fails in the rationals.