Book of Proof. (5 reviews). Richard Hammack, Virginia Commonwealth University . Pub Date: ISBN Publisher: Independent. Book of Proof has ratings and 11 reviews. David said: Playing with Numbers22 June – Sydney Well, what do you know, a university textbook that. This free book is an introduction to the language and standard proof methods of mathematics. – free book at

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There are a few things I would like to see dealt with in more depth–specifically, a few more examples and exercises for each of uniqueness proofs, “the following are equivalent” proofs, the Well-Ordering Principle, and strong induction. However, the choice and emphasis on most topics is highly satisfactory. The content contains standard proof techniques and results and, given its subject matter, is in no danger of becoming obsolete any time soon.

The approach in teaching the various proof outlines is especially relevant to novice proof-writers, particularly in Chapter 4 where illustrations show a proof being constructed, step by step, from the outline.

The text is written in a conversational tone that is easy for students to follow. New terms are always carefully defined, and a number of useful diagrams throughout the text add to the clarity of the explanations.

The text proceeds with one major topic per chapter suitable for discussing most chapters in one to two class periods. Each chapter further has a number of sections typically which make it easy to follow the book’s progression and to find relevant topics.

As it is a mathematics textbook, and particularly one on proof, notation and approaches to proofs adopted early in the text are used in the later chapters, but most readers will rarely if ever need to refer back to a previous chapter because of a reference in a later one. The organization of the text is one of its strengths. The book’s chapters are divided into four parts Fundamentals; How to Prove Conditional Statements; More on Proof; and Relations, Functions and Cardinalityand each part logically follows from the previous ones in a clear way.

The text is very easy to navigate, and there are no issues with the PDF files.

Book of Proof by Richard Hammack

One slight quirk is that the page numbers in the PDF file, due to introductory matter, are exactly 10 pages off from the page numbers appearing in the text, but it is easy to adapt to. The text is not culturally insensitive or offensive in any way. The examples used are from mathematics and largely devoid of references to any particular culture or background.

I have taught using this textbook in an introduction-to-proofs course over four semesters, and I am in general very satisfied with it. As mentioned previously, there are a few topics I feel the need to supplement when I teach, but on the whole I have felt very comfortable making this book an integral part of my course. I have used this book as the primary text for such a course twice, a course with two main goals: On the first role, the book really shines in its treatment of logic — sentences with quantifiers and their negations — methods of proof, induction basic and generalequivalence relations, functions, and cardinality.

Numerous examples are intertwined with introduction of concepts and thoughtful exercises echo the themes of each section. A high point is that the text ends with a rigorous treatment of the serious and magical results of Cantor on cardinality in addition to the Schroeder-Bernstein theorem.

Some instructors might see a lack of an introduction to delta-epsilon arguments as a weak point. Others might see the lack of delineation between logic and axiomatics as a weakness.

On the second role, the book lacks a sense of what the major might expect out of a mathematics degree and so when I use this book in a course I normally assign a cheap Dover secondary text for this purpose, along the lines of Ian Stewart’s “Concepts of Modern Mathematics,” the chapters of which naturally complement those of this text.

There is no shortage of such texts on the book market yet I don’t see myself changing this choice of text for my course anytime soon. The text also allows for a variety of pedagogical styles — with a nice mixture of good direct writing, examples, and a lot of relevant problems. The writing style of the text is best described as direct.


Students, who were expected to read considerable sections of the text before coming to class also reported that the text was very good and they liked that the price was right!

While almost every chapter depends on chapters preceding it there are pockets that I think are optional. I value the Euclidean algorithm and Bezout’s Theorem “the gcd of two integers can always be written as the integer linear combination of those two integers” and its corollaries but I don’t like the proof presented here and I think the topics can be held back until a course in number theory or in the opening weeks of abstract algebra.

Likewise, the perfect number theorem’s proof felt like a jump too high for many students so if time is pressing one could opt to postpone these topics in Chapters 7 and 8 respectively. A reversing of the order of Chapters 2 and 3 is also something I would recommend. The topics are presented in a clear fashion with themes in each section clearly stated and how one sections theme builds upon previously developed themes.

The online interface is a plain pdf that appears just as you would expect from the hardcopy. In the long run, the text might benefit from a mathjax-designed interface like that of, say, Judson’s “Abstract Algebra.

Book of Proof by Richard Hammack

For better or for worse, cultural relevance does not typically play into a mathematics text and this text is no different. I think this is a real shame — a richad we have paid collectively for emphasizing mathematics chiefly as a technocratic and scientific problem solving discipline as opposed to a humanistic and democratic problem framing one — but this is not a stick I wish to beat this text with, or at least this text alone.

As another reviewer pointed out, some of the problems in the logic section — negate “you can fool some people all the time, all This textbook covers an excellent choice of topics for an introductory course in mathematical proofs and reasoning. The book starts with the basics of set theory, logic and truth tables, and counting. Then, the book moves on to standard proof Then, the book moves on to standard proof techniques: These techniques will be useful in more advanced mathematics courses, as well as courses in statistics, computers science, and other areas.

The book ends with additional topics in relations, functions, and cardinality of sets. There is a preface, an introduction, an index, and solutions to selected exercises. While spending a few hours reading the book, I did not find any inaccuracies. The boik, theorems, and examples given, og well as the notation used, are good, standard, and well presented. For instance, I like how the book explains the differences among theorems, lemmas, corollaries, and propositions, since students sometimes are confused by such labels.

The material covered in this textbook is very relevant and fundamental in mathematics, and this book covers all of the main topics.

Relevance and longevity are not issue. The book is quite clear in explaining the various topics covered, particularly when it comes to set theory and basic proof techniques. I was impressed by how easy to read and well organized this textbook is.

Furthermore, the examples and figures are outstanding. The book is consistent in its use of definitions, diagrams, and terminology. Any redundancy, especially in terms of definitions, ot be useful rifhard preserve modularity. One could rearrange the order in which sections and topics in each chapter are covered, although it would be more challenging to rearrange chapters II, III, and IV rjchard covering chapter I first.

Also, mathematical induction could be covered before other proof techniques. The fundamentals of set theory, logic, and counting techniques are covered in chapter I. While mathematical induction could be covered before other proof techniques, it still works well to have it covered at the end of Chapter III. Relations, functions, and cardinality follow in chapter IV.

Interface is not an issue for this book. The diagrams, charts, boxes, tables, headings, and the use of boldface and italic font to indicate definitions and other key concepts, are very helpful to better organize the material.

One of my favorite diagrams is the one used to explain how mathematical induction works. One way to improve diagrams and figures would be to label all of them, to make them easier to refer to. There are no obvious grammatical errors, as far as I could see. This book is excellent for an introduction to mathematics proofs course.


Not only does it cover all of the main topics for such a course, but it also discusses mathematical writing, which is key when it comes to making mathematical concepts clear. Many students might know how to prove theorems or solve equations, but might not use correct mathematical notation.

The book is very useful to prepare students for courses such as advanced calculus, which is a proof-intensive course. The numerous examples and diagrams used are useful, not only to make the material easier to understand, but also to motivate students to learn more.

I would recommend this textbook to any instructor who teaches introduction to mathematical proofs, and to any student who is being exposed to this subject for the first time or needs to review this material. I use this book for a “Discrete Mathematics for Educators” course. Rchard students are all prospective middle and high school ov, and the main goals are to prepare prkof for upper level mathematics courses involving proofs, and to give them a The students are all prospective middle and high school teachers, and the main goals are to prepare them for upper level mathematics courses involving proofs, and to give them a brief introduction to discrete mathematics.

This fichard covers all of the needed proof techniques and gives interesting examples for them. I do use Ricard 3 combinatorics and add on some graph theory later on in the course. Thus, I would say it does a very nice job of both introducing students to proof and to intro number theory and combinatorics. Richagd of the content, this book passes the hamkack test.

We will not need to prepare students with introductions to other proof techniques except perhaps proof by computer? Very clear and well organized, and defines all new terminology. As a book used to transition students to upper level mathematics, this book does a very nice job of calling out mathematical language norms and writing norms.

The author provides a nice suggested organization at the beginning, but I have deviated a bit and this book is fine for that. I skipped the chapter on combinatorics and have not used those examples in the proofs so far.

After the first exam, we will do some combinatorics, and then go back and prove things about combinatorics and add in inductive proof techniques. The book’s structure definitely allows for these sorts of easy changes. I am deviating a bit this semester from the given order, but the book makes this easily doable, and it is still well organized even with the order mixed up a bit. I would love for hyperlinks to be added, so that you could click on the table of contents to get to chapters for example.

It is very easy to just do a search for terms to get there quickly, but this would be a great addition. I really enjoy this book and love that it is free for my students.

I’ve asked my students if they find the book useful and many have said they rely heavily on it. Also, since it is free I feel find going “off script” for a while – when I use an expensive text, I feel like I should make the most of the text for the students. But bc it is free I don’t feel that pressure. That said, I don’t find myself often deviating from the text’s content because it meets my needs. This text is intended for a transition or introduction to proof and proving in undergraduate mathematics.

Book of Proof – Open Textbook Library

Many of the elements needed for this transition are here, including predicate and propositional logic. The index is provided rrichard extensive. I have contacted the author about one typographical error I found during my reading, but it is error-free for the majority of the textbook. I love the content of this textbook. Since this topic is relevant for many aspiring mathematicians, the text will live a long life.