This is the final stretch on the road to the book being made available in physical form, hopefully in August.
Organisation of material
I have shuffled around some of the sections to increase the degree to which instructors are able to customise their courses. See below for how the new sections correspond with the sections in v0.1 and v0.2.
Most ‘introduction to proof’ classes cover, at the very least: logic, sets, functions, mathematical induction, and equivalence relations. The corresponding sections have now been pushed to the front of the book, in what now constitutes Part I: Core concepts.
Theoretically, a standalone course could be created that covers only Part I, although I do not believe that would be a fulfilling experience for students or instructors.
The remaining material now constitutes Part II: Topics, with chapters for number theory, enumerative combinatorics, real analysis, infinite sets and probability theory, and additional sections on order theory and structural induction.
Importantly, the book need not and should not be read from front to back; topics can be interspersed between the chapters or sections of Part I. Dependencies between sections are in the preface.
The most exciting development is that the (very) long-awaited sections on proof-writing in Appendix A have been written. These sections are intended to serve as a ‘companion’ to be read in parallel with the mathematical material in Parts I & II of the book.
A few previously unwritten or partially written sections have been written, including §7.2 Completeness and convergence, §7.3 Series and sums and §10.2 Structural induction.
New content on has been added to §8.1 Countable and uncountable sets — the reader is walked through a heuristic for proving a set is countable by showing its elements have finite descriptions over a countable alphabet.
End-of-chapter exercises have been added. Expanding these sections is one of my main tasks for the next month, so expect to see them grow as revisions are made to v0.3.
New sections have been added, notably §6.3 Alternating sums, which is devoted to proving combinatorial results involving—you guessed it—alternating sums. The inclusion–exclusion principle makes an appearance, but the focus is on using the involution principle to prove such results.
Ch 10 (Additional topics) contains a couple of sections that I’d like to evolve into chapters that I intend to write in future editions—I just don’t have enough time to get all of this done by August, so it will have to wait. But this is fun material nonetheless! Also in the pipeline for future versions of the book are chapters on abstract algebra, complex numbers, linear algebra, rings, and some basic topology.
Versions 0.2 and older were released under a CC BY-NC-SA 4.0 licence.
For v0.3, I have relaxed this to a CC BY-SA 4.0 licence. Relaxing the licence makes the book compatible with Wikipedia and the Stack Exchange network, for instance, so that material from the book can be reproduced on these websites (with proper attribution, of course).
The text size has been reduced from 11pt to 10pt. I never noticed how large it was until I printed out a couple of pages and cut them to size. This reduced the number of pages by about 12% (81 pages), which will ultimately reduce the cost of the print book when it is released.
Results, definitions, examples, exercises and tips/strategies are no longer distinguished only by name and colour, but now also by a symbol appearing in the margin. The colours have also been tweaked slightly to increase contrast. I made these changes to improve accessibility and make it easier to find what you’re looking for.
The TeX source for the book is now available on Github, and is released under the same licence as the book (CC BY-SA 4.0). Enjoy!
New organisation of material
|Ch 0 Getting started||Ch 0||§1.1|
|Ch 1 Logical structure||Ch 1||§1.2 and §2.1|
|Ch 2 Sets and functions||Ch 2||§2.2, §2.3, beginning of §4.1|
|Ch 3 Mathematical induction||§3.1||§1.3|
|Ch 4 Relations||§5.1||§5.1|
|Ch 5 Number theory||Ch 4||Ch 4|
|§6.1 Finite Sets||§3.2||End of §4.1 and beginning of §4.2|
|§6.2 Counting principles||§3.3||End of §4.2|
|§6.3 Alternating sums||—||—|
|Ch 7 Real analysis||Ch 7||Ch 6|
|§8.1 Countable and uncountable sets||§6.1||§4.3|
|§8.3 Cardinal arithmetic||§6.2||—|
|Ch 9 Probability theory||Ch 8||Ch 7|
|§10.1 Orders and lattices||§5.2||§5.2|
|§10.2 Structural induction||§5.3||§5.3|
|Apx A Mathematical writing||Apx A||Apx C incorporated as §A.3|
|Apx B Mathematical miscellany||Apx B||§B.2|
|Apx C Hints to selected exercises||Apx C||Apx A|
|Apx D Typesetting mathematics with LaTeX||§A.3||Apx C|