BOOK OF PROOF (Third Edition) Richard Hammack

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. Also see the Mathematical Association of America Math DL review (of the 1st edition), the Amazon reviews, and a brief news story on VCU InSight.

In this third edition, Chapter 3 on counting has been expanded and there is a new chapter on calculus proofs. Material has been edited for clarity, typos have been corrected, and some new exercises have been added. My decisions regarding revisions were guided by the Amazon reviews and many emails from readers.

The third edition will appear on Amazon and Barnes & Noble by October 2018. For now you can download a free PDF preview HERE. This version may undergo several minor tweaks before October. Please notify me if you find any typos!
•  Contents (Hover on the chapter title to see the subsections.)
•
•  Preface vii Introduction viii
• Part I: Fundamentals
• Part II: How to Prove Conditional Statements  4.   4.1    Theorems   4.2    Definitions   4.3    Direct Proof   4.4    Using Cases   4.5    Treating Similar Cases 113 5.   5.1    Contrapositive Proof   5.2    Congruence of Integers   5.3    Mathematical Writing 128 6.   6.1    Proving Statements with Contradiction   6.2    Proving Conditional Statements with Contradiction   6.3    Some Words of Advice 137
• Part III: More on Proof  7.   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.   8.1    How to Prove a ? A   8.2    How to Prove A ? B   8.3    How to Prove A = B   8.4    Examples: Perfect Numbers 157 9.   9.1    Disproving Universal Statements: Counterexamples   9.2    Disproving Existence Statements   9.3    Disproof by Contradiction 172 10.   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.   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.   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.   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.   14.1    Sets With Equal Cardinality   14.2    Countable and Uncountable Sets   14.3    Comparing Cardinalities   14.4    The Cantor-Bernstein-Schröder Theorem 269
 Conclusion 291 Solutions 292 Index 365
Notice: The Creative Commons License allows you to freely use or share the PDF version of the book, in full or in part, provided you acknowledge it as the Author's work. You do not have license to alter the PDF in any way, nor may you sell it or use it for commercial purposes.

Note to adopters: Your bookstore will need to know that the book will be distributed by Ingram. Please let me know if you use Book of Proof in your classes. I maintain a list of known adoptions here. Thanks!