A TeXas Style Introduction to Proof Vol 48 by Ron Taylor, ISBN-13: 978-1470450465

A TeXas Style Introduction to Proof Vol 48 by Ron Taylor, ISBN-13: 978-1470450465

[PDF eBook eTextbook]

• Publisher: ‎ American Mathematical Society (July 26, 2019)
• Language: ‎ English
• 161 pages
• ISBN-10: ‎ 1470450461
• ISBN-13: ‎ 978-1470450465

A TeXas Style Introduction to Proof is an IBL textbook designed for a one-semester course on proofs (the bridge course ) that also introduces TeX as a tool students can use to communicate their work. As befitting textless text, the book is, as one reviewer characterized it, minimal. Written in an easy-going style, the exposition is just enough to support the activities, and it is clear, concise, and effective. The book is well organized and contains ample carefully selected exercises that are varied, interesting, and probing, without being discouragingly difficult.

Table of Contents:

Acknowledgements xiii
0 Introduction 1
0.1 To the instructor . . . . . . . . . . . . . . . . . . . . . 2
0.2 To the student . . . . . . . . . . . . . . . . . . . . . . 2
0.3 How to construct those proofs . . . . . . . . . . . . . 4
0.4 Using LATEX to write mathematics . . . . . . . . . . . 8
0.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . 11
0.6 The journey begins . . . . . . . . . . . . . . . . . . . . 12
1 Symbolic logic 15
1.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2 Compound statements and logical connectives . . . . . 18
1.3 Proof via truth table . . . . . . . . . . . . . . . . . . . 22
1.4 Implications . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6 Compound quantifiers . . . . . . . . . . . . . . . . . . 31
2 Proof methods 35
2.1 Variable names . . . . . . . . . . . . . . . . . . . . . 40
2.2 Parity and divisibility . . . . . . . . . . . . . . . . . . 41
2.3 Negations . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Proof methods . . . . . . . . . . . . . . . . . . . . . . 48
3 Mathematical induction 49
3.1 Geometric tilings . . . . . . . . . . . . . . . . . . . . 49
3.2 Induction versus deduction . . . . . . . . . . . . . . . 53
3.3 Strong Induction . . . . . . . . . . . . . . . . . . . . 61
4 Set theory 63
4.1 Notation and definitions . . . . . . . . . . . . . . . . . 64
4.2 Venn diagrams . . . . . . . . . . . . . . . . . . . . . . 70
4.3 General proofs with sets . . . . . . . . . . . . . . . . . 72
4.4 Set operations . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Deeper thinking . . . . . . . . . . . . . . . . . . . . . 77
4.6 Set products . . . . . . . . . . . . . . . . . . . . . . . 78
4.7 Power sets . . . . . . . . . . . . . . . . . . . . . . . . 80
4.8 Index sets and set operations . . . . . . . . . . . . . . 81
4.9 Spaciousness . . . . . . . . . . . . . . . . . . . . . . 86
5 Functions and relations 87
5.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Partitions . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3 Order relations . . . . . . . . . . . . . . . . . . . . . 98
5.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5 Throwing some math around . . . . . . . . . . . . . . 102
6 Counting 109
6.1 A (very) brief history of infinity . . . . . . . . . . . . 109
6.2 Finite sets . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3 The Pigeonhole Principle . . . . . . . . . . . . . . . . 112
6.4 A foundation for infinity . . . . . . . . . . . . . . . . 114
6.5 Can we go beyond infinity? . . . . . . . . . . . . . . . 118
7 Axiomatics 123
7.1 LSAT axiomatics . . . . . . . . . . . . . . . . . . . . 124
7.2 Charles Dodgson’s axiomatic looking-glass . . . . . . 128
7.3 Shiny hidden people . . . . . . . . . . . . . . . . . . . 128
A Mathematical writing 131
B Comments on Style 133
C The Structure of a LATEX Document 135
C.1 A sample LATEX document . . . . . . . . . . . . . . . 136
C.2 The Preamble . . . . . . . . . . . . . . . . . . . . . . 136
C.3 The Text . . . . . . . . . . . . . . . . . . . . . . . . . 138
C.4 Formatting text . . . . . . . . . . . . . . . . . . . . . 139
C.5 Typesetting mathematics . . . . . . . . . . . . . . . . 140
C.6 LATEX codes for common mathematical symbols . . . . 141
C.7 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C.8 Arrays with reasons . . . . . . . . . . . . . . . . . . . 146
C.9 Making lists (Checking them twice is a good idea.) . . . . 148
C.10 An example of a homework assignment in LATEX . . . . 149
C.11 TEX Source Code for the example . . . . . . . . . . . 149
Index 155
About the Authors 161

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