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Book of Proof

Richard Hammack
Creative Commons
Publication Date: 
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[Reviewed by
David Offner
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This is a wonderful book. Written as a text for a one-semester “transition to higher mathematics” course, it introduces the undergraduate to logic and proofs and to the basic objects and language used in higher mathematics. It is ideal for the many American undergraduates who come to college with little or no experience with proof or formal reasoning and need to be brought up to speed quickly in order to succeed in upper-level mathematics courses.

The book is divided into four parts. The first part introduces sets and logic and includes a chapter on counting, which gives a brief introduction to elementary combinatorial techniques. The second section covers the basic techniques for proving conditional statements: direct proof, contrapositive proof, and proof by contradiction. The third part provides more examples of common proofs, such as proving non-conditional statements, proofs involving sets, and disproving statements, and also introduces mathematical induction. Finally the fourth part returns to basic mathematical structures, discussing relations, functions, and cardinality.

Hammack clearly knows his audience, and writes about precise mathematical ideas in an inviting, conversational style that can be read by mathematically immature undergraduates. The exposition has the tone of a friendly office hour chat, as Hammack frequently addresses the reader as “you,” (e.g. “You should examine the following statements and make sure you understand how the answers were obtained.” p. 15) and sprinkles the text with references to what the reader has encountered in previous math courses, or will encounter in the future.

The book contains many well-chosen examples, non-examples, and exercises that illustrate key concepts and clarify common confusions. The deliberate pace of the book allows Hammack to point out many of these explicitly, and give helpful advice about how to think about them (e.g. “You can think of statements as pieces of information that are either correct or incorrect.” p. 32). In explaining proof techniques or types of proofs, he gives helpful templates, and very nice discussions of not only the logic of proofs, but how one goes about constructing them in practice.

Besides giving students the tools required to pursue advanced mathematics, the book also provides a nice introduction to the culture of mathematics. Engaging mathematical topics are sprinkled throughout. Many great theorems and proofs arise naturally in such a text: the ancient proofs that the square root of 2 is irrational and that there are an infinite number of primes are examples. Other engaging topics, such as Russell’s paradox, perfect numbers, Fibonacci numbers, and infinite cardinalities, are found in sections that shoot off from the main stream of the text.

One last attraction of Book of Proof is its price, or lack thereof. It is freely available for download from the author’s website as a .pdf document, and is also available as a print on demand paperback at for $12.95.

There are many places where the author makes choices between rigor and accessibility, for example in deciding what properties of the integers should be assumed and what should be proved. While the book generally makes excellent choices in this regard, there are some proofs that do not give a significant pay-off given the investment. For example, much effort is put into building up to a proof of unique prime factorization of the integers which comes late in the book. While this is a spectacular result, many of the intermediate steps along the way are not well motivated as they arise, and may not be appreciated by students. This is surely a matter of taste, however, and much will depend on the particular students involved. It will probably come as no surprise that it is possible to skip certain sections of the book with no complaint from students.

I have used the book as a text in a sophomore-level discrete mathematics course, where we have covered most but not all of the material in the book, and supplemented the text with additional topics in discrete mathematics. My students and I have both loved the book. They find it readable and interesting, and they learn enough from reading the book that I rarely need to explain any topic from scratch. I highly recommend this book as a text or supplement for an undergraduate transition course. Furthermore, it is readable and enjoyable enough that it would be interesting and useful independent reading for almost any undergraduate mathematics major.

David Offner is Assistant Professor of Mathematics at Westminster College in New Wilmington, PA.



I: Fundamentals
1. Sets
2. Logic
3. Counting

II: How to Prove Conditional Statements
4. Direct Proof
5. Contrapositive Proof
6. Proof by Contradiction

III: More on Proof
7. Proving Non-Conditional Statements
8. Proofs Involving Sets
9. Disproof
10. Mathematical Induction

IV: Relations, Functions and Cardinality
11. Relations
12. Functions
13. Cardinality