# Book of Proof - Third Edition

Richard Hammack, Virginia Commonwealth University

Copyright Year: 2013

ISBN 13: 9780989472104

Publisher: Richard Hammack

Language: English

## Conditions of Use

Attribution-NoDerivs

CC BY-ND

## Reviews

This textbook is very comprehensive. Covers a basic review of sets and set operations, logic and logical statements, all the proof techniques, set theory proofs, relation and functions, and additional material that is helpful for upper-level proof... read more

This textbook is very comprehensive. Covers a basic review of sets and set operations, logic and logical statements, all the proof techniques, set theory proofs, relation and functions, and additional material that is helpful for upper-level proof course preparation (like a chapter on proofs in calculus). There is plenty of material in the book for a very thorough treatment of proofs and flexibility with other chapters devoted to counting, calculus, and other material.

I have not found any errors in the textbook other than a place where the author says he is using a proof of the contrapositive but proceeds to prove it by a proof by contradiction.

The textbook is very relevant and has served as a good textbook for an introduction to proof course. Adding some more homework problems that present a new mathematical definition and corresponding proofs dealing with this new definition would be very beneficial for introduction to proof students (to prepare students for aspects in upper-level proof courses).

The clarity of the book is very good. I just have one minor thing that would help with clarity. I stress at the beginning of the class the mathematical norms of the class (what we expect in terms of proofs and proving in the class for the semester) and the book points out that proofs are written in paragraph form with complete sentence. However, the book contradicts this throughout the book with proofs that do not follow this format.

The textbook is very consistent except on the aspect listed in the previous comment about paragraph proofs.

The way the book is organized, things are easily divided into smaller sections. There is a few longer chapters, but these chapters are divided in to manageable sections. Usually I can cover at least one section or two for each class.

The author does a great job on organizing the book. The only chapter that I don't usually cover is Chapter 3 and the Chapter on proofs in Calculus.

I love the downloaded pdf with quick links to particular Chapter and Sections in the book. I have not had an issues with this book.

I have not observed any grammatical errors in the text.

I haven't found any text in the book that is culturally insensitive or offensive.

I would definitely recommend this textbook to other colleagues.

This book covers all of the major areas of a standard introductory course on mathematical rigor/proof, such as logic (including truth tables) proof techniques (including contrapositive proof, proof by contradiction, mathematical induction, etc.),... read more

This book covers all of the major areas of a standard introductory course on mathematical rigor/proof, such as logic (including truth tables) proof techniques (including contrapositive proof, proof by contradiction, mathematical induction, etc.), and fundamental notions of relations, functions, and set cardinality (ending with the Schroder-Bernstein Theorem). 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 is accurate and error-free.

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 book is very internally consistent, in both approach and pacing.

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 3-5) 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 Cardinality), and 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. 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.

There are no known grammatical errors.

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.

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 course with two main goals: prepare the student for proof-centric classes like abstract algebra and... read more

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 course with two main goals: prepare the student for proof-centric classes like abstract algebra and real analysis, and introduce the student to what the major ought to look like and what mathematics hopes to achieve beyond the calculus. 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 general), equivalence 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.

Errors are rare, content is accurate.

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!).

The book is consistent in terms of terminology.

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 text is available online (for free) or for hardcopy purchase (~$15) and the two versions line up. 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."

No grammatical errors.

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 price 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...." -- are perhaps a bit unusual but my students and I appreciated the ambiguity present in the real world and these problems presented an opportunity to contrast mathematical definitions with the ambiguity of language more generally.

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... read more

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 techniques: direct proof, proof by contrapositive and contradiction, proving existence and uniqueness, constructive proof, proof by induction, and others. 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 definitions, theorems, and examples given, as 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, can be useful to preserve modularity.

This book’s modularity is good, for an introduction to proofs course. 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 without covering chapter I first. Chapter IV could be covered before chapters II and III. Also, mathematical induction could be covered before other proof techniques.

Even though certain sections and topics could be rearranged, given the textbook’s modularity, I think that the order in which the topics are covered is very logical. The fundamentals of set theory, logic, and counting techniques are covered in chapter I. These concepts are needed in order to cover proof techniques in chapters II and III. 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. I would only suggest to avoid the use of apostrophes in expressions such as “it’s” or “we’ve”. Instead, write “it is” or “we have”, as these expressions are better suited for professional writing.

Cultural relevance is not an issue for this textbook.

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. 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... read more

I use this book for a "Discrete Mathematics for Educators" course. 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 book covers all of the needed proof techniques and gives interesting examples for them. I do use Chapter 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.

After two semesters of teaching from it I have not found an error.

Because of the content, this book passes the longevity test. We will not need to prepare students with introductions to other proof techniques (except perhaps proof by computer?), though additional introductory discrete topics would be great additions for me, though I am using the text for a specific course that goes beyond the scope of the book's intentions.

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.

Very consistent.

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.

Very well organized. I especially like the advanced organizer that provides suggested exam timing w/r/t the chapters. 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.

No issues here. 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.

No errors.

Not applicable? So - it is vacuously highly culturally relevant.

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. Many of the elements needed for this transition are here, including predicate and propositional logic. The index is provided and extensive. read more

This text is intended for a transition or introduction to proof and proving in undergraduate mathematics. Many of the elements needed for this transition are here, including predicate and propositional logic. The index is provided and 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.

I believe that the clarity of the text is wonderful, with the exception of one section. I thought that section 5.3, "Mathematical Writing," could have been worded a bit differently or presented to the reader with more discussion. It seemed like I was scolded during the section. I understand the intention of the section, and I praise the author for putting in a section like this since most mathematics textbooks do not, but it seemed to be "good or bad" instead of what the author stated, "good or bad writing is sometimes a matter of opinion." Perhaps the author could include some statements to have the reader read, like run-on sentences in english, and determine these rules for his/herself.

The text is consistent in framework and terminology. I found no discrepancies while reading.

There are certain unavoidable parts of the text that are self-referential. This is common with mathematics. I do enjoy that many sections of the text are about techniques of proving, which highlight the technique's importance in mathematics.

I really enjoyed the topics in the text, including Chapters 4-10. The proving techniques are the cornerstones to mathematics. I will be incorporating these sections in my future courses due to the elegant way the author has handled these techniques. This would get a higher rating if I could.

The interface was fine. It was a PDF version of a proof textbook. What I wanted from the text was to somehow incorporate the additional nuances that make eBooks slightly better than paper. For example, in the index, I would have liked hyperlinks back to the page where the term was defined. Perhaps I am being critical, but I think that eBooks or PDF versions should have these small refinements.

This text contains no grammatical errors from what I've observed.

There is one question for which I contacted the author on. On page 47, after the section on statements and truth of statements, there is an exercise to determine if a sentence is a statement, and if it is, determine if that statement is true. The statement is #15: "In the beginning, God created the heaven and the earth." In my opinion, this question may lead to discussions that stray from the original task. Other than this, I found the text to be not culturally insensitive or offensive.

Although there are certain aspects of the text I would modify, these aspects are minute compared to the amount of understanding and depth that the author introduces. The author has clearly taken his time in developing a textbook that can be accessible and transferrable to subsequent courses. I highly recommend any transition-to-proof or introduction-to-proof course be taught with this textbook.

## Table of Contents

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. Proofs in Calculus
- 14. Cardinality of Sets

## Ancillary Material

## 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

### Author

**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.