© 2018 by Richard Hammack; Licensed under a Creative Commons Attribution-NonCommercial-NoDerivative 4.0 License
 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 Paradox 3 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 Note 34 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 Proof 65
• 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 Cases 113 5.1    Contrapositive Proof   5.2    Congruence of Integers   5.3    Mathematical Writing 128 6.1    Proving Statements with Contradiction   6.2    Proving Conditional Statements with Contradiction   6.3    Combining Techniques   6.4    Some Words of Advice 137
• 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 Proofs 147 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 Numbers 157 9.1    Disproving Universal Statements: Counterexamples   9.2    Disproving Existence Statements   9.3    Disproof by Contradiction 172 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 Numbers 180
• 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 Sets 201 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 Preimage 223 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    Series 244 14.1    Sets With Equal Cardinality   14.2    Countable and Uncountable Sets   14.3    Comparing Cardinalities   14.4    The Cantor-Bernstein-Schröder Theorem 269
 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 rhammack@vcu.edu 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).