Proofs: A Long-Form Mathematics Textbook by Jay Cummings, ISBN-13: 979-8595265973
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- Publisher: Independently published (January 19, 2021)
- Language: English
- 511 pages
- ISBN-10: 8595265976
- ISBN-13: 979-8595265973
This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by “scratch work” or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own.This book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. The text aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and conversational, and includes periodic attempts at humor.This text is also an introduction to higher mathematics. This is done in-part through the chosen examples and theorems. Furthermore, following every chapter is an introduction to an area of math. These include Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality and group theory.After every chapter are “pro-tips,” which are short thoughts on things I wish I had known when I took my intro-to-proofs class. They include finer comments on the material, study tips, historical notes, comments on mathematical culture, and more. Also, after each chapter’s exercises is an introduction to an unsolved problem in mathematics. In the first appendix we discuss some further proof methods, the second appendix is a collection of particularly beautiful proofs, and the third is some writing advice.
Table of Contents:
1 Intuitive Proofs 1
1.1 Chessboard Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Naming Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Bonus Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Direct Proofs 35
2.1 Working From Definitions . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Proofs by Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . 46
2.5 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 Bonus Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Sets 73
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2 Proving A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Proving A = B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5 Two Final Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.6 Bonus Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Induction 107
4.1 Dominoes, Ladders and Chips . . . . . . . . . . . . . . . . . . . . . . 107
4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3 Strong Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.4 Non-Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.5 Bonus Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5 Logic 155
5.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.2 Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.3 Quantifiers and Negations . . . . . . . . . . . . . . . . . . . . . . . . 167
5.4 Proving Quantified Statements . . . . . . . . . . . . . . . . . . . . . 174
5.5 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.6 Bonus Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6 The Contrapositive 197
6.1 Finding the Contrapositive of a Statement . . . . . . . . . . . . . . . 199
6.2 Proofs Using the Contrapositive . . . . . . . . . . . . . . . . . . . . . 200
6.3 Bonus Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7 Contradiction 213
7.1 Two Warm-Up Examples . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7.3 The Most Famous Proof in History . . . . . . . . . . . . . . . . . . . 219
7.4 The Pythagoreans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.5 Bonus Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Introduction to Game Theory . . . . . . . . . . . . . . . . . . . . . . 241
8 Functions 247
8.1 Approaching Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.2 Injections, Surjections and Bijections . . . . . . . . . . . . . . . . . . 251
8.3 The Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
8.4 Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
8.5 Bonus Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
Introduction to Cardinality . . . . . . . . . . . . . . . . . . . . . . . . 283
9 Relations 291
9.1 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9.2 Abstraction and Generalization . . . . . . . . . . . . . . . . . . . . . 303
9.3 Bonus Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Introduction to Group Theory . . . . . . . . . . . . . . . . . . . . . . 319
Jay Cummings believes that learning math has become far too expensive, and is striving to write textbooks which are enjoyable to read, highlight the beauty in mathematics, and are significantly more affordable than the others on the market. After receiving his PhD from UC San Diego under Ron Graham, he joined the faculty at California State University, Sacramento. His combinatorics research continues and he particularly enjoys involving students in his work. He is passionate about his teaching and enjoys designing new courses.
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