An Infinite Descent into Pure Mathematics.pdf

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Note to readers

Contents

Acknowledgements

1 Mathematical reasoning

1.1 Getting started

1.2 Elementary proof techniques

1.3 Induction on the natural numbers

2 Logic, sets and functions

2.1 Symbolic logic

2.2 Sets and set operations

2.3 Functions

3 Number theory

3.1 Division

3.2 Prime numbers

3.3 Modular arithmetic

4 Finite and infinite sets

4.1 Functions revisited

4.2 Counting principles

4.3 Infinite sets

5 Relations

5.1 Relations

5.2 Orders and lattices

5.3 Well-foundedness and structural induction

6 Real analysis

6.1 Inequalities and bounds

6.2 Sequences and convergence

6.3 Series and sums

7 Discrete probability theory

7.1 Discrete probability spaces

7.2 Discrete random variables

7.3 Expectation

8 Additional topics

8.1 Ring theory

8.2 Ordinal and cardinal numbers

8.3 Boolean algebra

8.4 Complex numbers

8.5 Limits and asymptotes

A Hints for selected exercises

B Foundations

B.1 Logical theories and models

B.2 Set theoretic foundations

B.3 Other foundational matters

C Typesetting mathematics in LaTeX

Index

An inﬁnite descent into pure mathematics (John Mackey’s forked version)∞∞∞∞ ∞ ∞ ∞ ... by Clive Newstead Last updated on Monday 26th November 2018

2 2

Note to readers Hello, and thank you for taking the time to read this quick introduction to the book! I would like to begin with an apology and a warning: This book is still under development! That is to say, there are some sections that are incomplete (notably Sections 6.2 and 6.3, and all of Chapter 8), as well as other sections which are currently much more terse than I would like them to be. As the book is undergoing constant changes, I advise that you do not print the notes in their entirety—if you must print them at all, then I suggest that you do it a few pages at a time, as required. One book, two versions This is a forked version of Clive’s book, being maintained independently by John Mackey, who might add, remove or change content at his discretion, and is hosting it on his own web page. The original version of this book is available for download from the following web page: http://infinitedescent.xyz John’s version was forked as of 25th May 2018. Please be aware that these two versions may eventually diﬀer in essential ways! 3

4 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 3.2 3.3 8.1 4.1 8.4 6.1 5.1 4.2 7.1 6.2 5.2 4.3 7.2 6.3 5.3 8.2 7.3 8.5 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 , 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.) 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. 4

5 Licence This book is licensed under the Creative Commons Attribution-NonCommercial- ShareAlike 4.0 (cc by-nc-sa 4.0) licence. This means you’re welcome to share this book, provided that you give credit to the author, and that any copies or derivatives of this book are released under the same licence, are freely available and are not for commercial use. The full licence is available at the following link: http://creativecommons.org/licenses/by-nc-sa/4.0/ Comments and corrections Any feedback, be it from students, teaching assistants, instructors or any other readers, would be very much appreciated. Particularly useful are corrections of typographical errors, suggestions for alternative ways to describe concepts or prove theorems, and requests for new content (e.g. if you know of a nice example that illustrates a concept, or if there is a relevant concept you wish were included in the book). Such feedback can be sent to me by email (cnewstead@cmu.edu). 5

6 6

Contents Note to readers Acknowledgements 1 Mathematical reasoning 1.1 Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Elementary proof techniques . . . . . . . . . . . . . . . . . . . . . . . 1.3 Induction on the natural numbers . . . . . . . . . . . . . . . . . . . . 2 Logic, sets and functions 3 11 13 14 32 51 79 2.1 Symbolic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.2 Sets and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3 Number theory 129 3.1 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.2 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.3 Modular arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7

8 4 Finite and inﬁnite sets Contents 177 4.1 Functions revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.2 Counting principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 4.3 Inﬁnite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 5 Relations 231 5.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 5.2 Orders and lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 5.3 Well-foundedness and structural induction . . . . . . . . . . . . . . . 255 6 Real analysis 267 6.1 Inequalities and bounds . . . . . . . . . . . . . . . . . . . . . . . . . 268 6.2 Sequences and convergence . . . . . . . . . . . . . . . . . . . . . . . 293 6.3 Series and sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 7 Discrete probability theory 309 7.1 Discrete probability spaces . . . . . . . . . . . . . . . . . . . . . . . . 310 7.2 Discrete random variables . . . . . . . . . . . . . . . . . . . . . . . . 329 7.3 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 8 Additional topics 347 8.1 Ring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 8.2 Ordinal and cardinal numbers . . . . . . . . . . . . . . . . . . . . . . 354 8.3 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 8.4 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 8.5 Limits and asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . 357 A Hints for selected exercises 359 8

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