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Navigating the book
The material covered in Chapters 1 and 2 can be considered prerequisite for all sub-
sequent material in the book; any introductory course in pure mathematics should
cover at least these two chapters. The remaining chapters are a preview of other
areas of pure mathematics. The dependencies between the sections in Chapters 3–8;
dashed arrows indicate that a section is a recommended, rather than required, for
another.
3.1 4.1 4.2 4.3 8.2
3.2 8.4 7.1 7.2 7.3
3.3 6.1 6.2 6.3 8.5
8.1 5.1 5.2 5.3 8.3
What the numbers, colours and symbols mean
Much of the material in this book is broken into enumerated items which, broadly
speaking, fall into one of four categories: results (often followed by proofs), defin-
itions, examples (including exercises for the reader), and remarks. These items
are colour-coded as indicated in the previous sentence, and are enumerated accord-
ing to their section—for example, Theorem 1.3.10 is in Section 1.3. Particularly
important theorems, definitions and so on, appear in a box .
You will also encounter the symbols , C and ~, whose meanings are as follows:
End of proof. It is standard in mathematical documents to identify when a
proof has ended by drawing a small square or by writing ‘Q.E.D.’ (The latter
stands for quod erat demonstrandum, which is Latin for what was to be shown.)
C End of item. This is not a standard usage, and is included only to help
you to identify when an item has finished and the main content of the book
continues.
~ Optional content. Sections, exercises, results and proofs marked with this
symbol can be skipped over. Usually this is because the content is very chal-
lenging, or is technical in a way that is mathematically necessary but educa-
tionally not very important.
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