Book of Proof

(4 reviews)


Richard Hammack, Virginia Commonwealth University

Pub Date: 2013

ISBN 13: 978-0-9894721-0-4

Publisher: Independent

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Reviewed by Milos Savic, Assistant Professor, University of Oklahoma, on 1/13/2015.

This text is intended for a transition or introduction to proof and proving in undergraduate mathematics. Many of the elements needed for this … read more



Reviewed by Jess Ellis, Assistant Professor, Colorado State University, Fort Collins, on 1/8/2016.

I use this book for a "Discrete Mathematics for Educators" course. The students are all prospective middle and high school teachers, and the main … read more



Reviewed by Roberto Munoz-Alicea, Instructor/Academic Support Coordinator, Colorado State University, on 1/8/2016.

This textbook covers an excellent choice of topics for an introductory course in mathematical proofs and reasoning. The book starts with the basics … read more



Reviewed by Edwin O'Shea, Associate Professor, James Madison University, on 4/12/2017.

The text is very suitable for an "introduction to proofs/transitions" course. I have used this book as the primary text for such a course twice, a … read more


Table of Contents

  • Chapter 1: Fundamentals
  • Chapter 2: How to Prove Conditional Statements
  • Chapter 3: More on Proof
  • Chapter 4: Relations, Functions and Cardinality

About the Book

This is a book about how to prove theorems.

Until this point in your education, you may have regarded mathematics primarily as a computational discipline. You have learned to solve equations, compute derivatives and integrals, multiply matrices and find determinants; and you have seen how these things can answer practical questions about the real world. In this setting, your primary goal in using mathematics has been to compute answers.

But there is another approach to mathematics that is more theoretical than computational. In this approach, the primary goal is to understand mathematical structures, to prove mathematical statements, and even to invent or discover new mathematical theorems and theories. The mathematical techniques and procedures that you have learned and used up until now have their origins in this theoretical side of mathematics. For example, in computing the area under a curve, you use the fundamental theorem of calculus. It is because this theorem is true that your answer is correct. However, in your calculus class you were probably far more concerned with how that theorem could be applied than in understanding why it is true. But how do we know it is true? How can we convince ourselves or others of its validity? Questions of this nature belong to the theoretical realm of mathematics. This book is an introduction to that realm.

This book will initiate you into an esoteric world. You will learn and apply the methods of thought that mathematicians use to verify theorems, explore mathematical truth and create new mathematical theories. This will prepare you for advanced mathematics courses, for you will be better able to understand proofs, write your own proofs and think critically and inquisitively about mathematics.

This text has been used in classes at: Virginia Commonwealth University, Lebanon Valley College, University of California - San Diego, Colorado State University, Westminster College, South Dakota State University, PTEK College - Brunei, Christian Brothers High School, University of Texas Pan American, Schola Europaea, James Madison University, Heriot-Watt University, Prince of Songkla University, Queen Mary University of London, University of Nevada - Reno, University of Georgia - Athens, Saint Peter's University, California State University, Bogaziçi University, Pennsylvania State University, University of Notre Dame

About the Contributors


Richard Hammack, PhD is an Associate Professor in the Department of Mathematics and Applied Mathematics at Virginia Commonwealth University. He received his PhD in mathematics from the University of North Carolina at Chapel Hill.