Preface

Every definition of analysis is a sentence about sets and functions. Logic and Proof supplied the grammar of such sentences: the connectives, the quantifiers, and the prose that carries their rules. This volume supplies the nouns: sets and functions, relations and orders, the natural numbers and induction, and the sizes of infinite sets.

Who it is for. The reader we have in mind has worked through Logic and Proof, or has its habits from elsewhere: reading a statement with its quantifiers in place, proving a “for every” by taking an arbitrary element, naming a witness before using it. The chapters point back to Logic and Proof where they use one of its rules, so a reader who is unsure of a step can look it up.

What it covers.

Chapter	What it does
1	Sets and the working rules that say which sets exist, Russell’s paradox, families of sets, products and relations, functions and their inverses, images and preimages
2	The arithmetic of $\mathbb{N}$ and $\mathbb{Z}$ from induction, well-ordering, induction as a craft, definition by recursion, equivalence relations and partitions, partial orders
3	Finite and countable sets, the axiom of choice, diagonal arguments and Cantor’s theorem, the Schröder–Bernstein theorem, infinite sets; and Gate 0
4	Zorn’s lemma

Chapters 1–3 are written; Chapter 4 is not yet. The real numbers are used throughout as familiar objects, with their school arithmetic and order, and every use of a school fact is named where it occurs. Analysis on the Real Line turns those facts into consequences of axioms, and Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ builds the numbers out of sets.

How to read it. Read the chapters in order: each uses the ones before it. Every section has Quick checks with folded answers; answer each one before you unfold it. The exercises of a section come in three groups, A to check understanding, B for practice and C to go deeper, and the last section of each chapter adds R items that reach back to earlier chapters. A chapter’s opening page lists what it needs, with links, and the time it takes.

Gate 0. Chapter 3 ends with an examination on Logic and Proof and the first three chapters of this volume: a definition drill and six problems, closed book, with a marking scheme and a map that sends each lost problem back to named sections. Pass it before starting Analysis on the Real Line. Chapter 4 is not needed there.