BOOK OF PROOF | Third Edition | Richard Hammack |
Paperback: ISBN: 978-0-9894721-2-8 ($21.75) | Hardcover: ISBN: 978-0-9894721-3-5 ($36.15) |
This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative. See other endorsements here. An adoptions list is
here, and ancillary materials are here. See also the Translations Page.
You can order a copy through Barnes & Noble or Amazon. You can also download a free PDF version HERE. (The contents links below will take you to specific chapters in this file.)
Contents
(Hover on the chapter title to see the subsections.)
-
Preface
vii
Introduction viii
- Part I: Fundamentals
1.1 Introduction to Sets
1.2 The Cartesian Product
1.3 Subsets
1.4 Power Sets
1.5 Union, Intersection, Difference
1.6 Complement
1.7 Venn Diagrams
1.8 Indexed Sets
1.9 Sets That Are Number Systems
1.10 Russel's Paradox3
2.1 Statements
2.2 And, Or, Not
2.3 Conditional Statements
2.4 Biconditional Statements
2.5 Truth Tables for Statements
2.6 Logical Equivalence
2.7 Quantifiers
2.8 More on Conditional Statements
2.9 Translating English to Symbolic Logic
2.10 Negating Statements
2.11 Logical Inference
2.12 An Important Note34
3.1 Lists
3.2 The Multiplication Principle
3.3 The Addition and Subtraction Principles
3.4 Factorials and Permutations
3.5 Counting Subsets
3.6 Pascal's Triangle and the Binomial Theorem
3.7 The Inclusion-Exclusion Principle
3.8 Counting Multisets
3.9 The Division and Pigeonhole Principles
3.10 Combinatorial Proof65
- Part II: How to Prove Conditional Statements
4.1 Theorems
4.2 Definitions
4.3 Direct Proof
4.4 Using Cases
4.5 Treating Similar Cases113
5.1 Contrapositive Proof
5.2 Congruence of Integers
5.3 Mathematical Writing128
6.1 Proving Statements with Contradiction
6.2 Proving Conditional Statements with Contradiction
6.3 Combining Techniques
6.4 Some Words of Advice137
- Part III: More on Proof
7.1 If-And-Only-If Proof
7.2 Equivalent Statements
7.3 Existence Proofs; Existence and Uniqueness Proofs
7.4 Constructive Versus Non-Constructive Proofs147
8.1 How to Prove a is an element of A
8.2 How to Prove A is a subset of B
8.3 How to Prove A = B
8.4 Examples: Perfect Numbers157
9.1 Disproving Universal Statements: Counterexamples
9.2 Disproving Existence Statements
9.3 Disproof by Contradiction172
10.1 Proof by Induction
10.2 Proof by Strong Induction
10.3 Proof by Smallest Counterexample
10.4 Examples: The Fundamental Theorem of Arithmetic
10.5 Fibonacci Numbers180
- Part IV: Relations, Functions and Cardinality
11.1 Relations
11.2 Properties of Relations
11.3 Equivalence Relations
11.4 Equivalence Classes and Partitions
11.5 The Integers Modulo n
11.6 Relations Between Sets201
12.1 Functions
12.2 Injective and Surjective Functions
12.3 The Pigeonhole Principle Revisited
12.4 Composition
12.5 Inverse Functions
12.6 Image and Preimage223
13.1 The Triangle Inequality
13.2 Definition of a Limit
13.3 Limits That Do Not Exist
13.4 Limit Laws
13.5 Continuity and Derivatives
13.6 Limits at Infinity
13.7 Sequences
13.8 Series244
14.1 Sets With Equal Cardinality
14.2 Countable and Uncountable Sets
14.3 Comparing Cardinalities
14.4 The Cantor-Bernstein-Schr�der Theorem269
Solutions 292
Index 365
Thanks to the readers who wrote to report mistakes and typos! I incorporate reader feedback in periodic revisions. Please contact me at if you find any additional mistakes, no matter how minor.
Notice: On June 14, 2022 I issued edition 3.3 in print and PDF (ISBN's unchanged). This slight revision corrects a handful of typos found by readers. All orders printed after June 14 will be edition 3.3.
Note to adopters: Your bookstore will need to know that the book is distributed by Ingram. Please let me know if you use Book of Proof in your classes and I will update the adoptions list. Thanks!
Notice: The Creative
Commons License allows you to freely use or share the book's PDF, in full or in part, provided you acknowledge it as the Author's work. It does not permit altering of content for anything other than personal use. Commercial use is forbidden (except by the publisher).