Book of Proof - Third Edition

Reviewed by Kenneth Levasseur, Professor, University of Massachusetts Lowell on 5/29/24

Comprehensiveness rating: 5 see less

This book could serve as a complete text for an introductory course on mathematical proof. It includes all of the important topics that student need to know to study upper-level mathematics courses such as real analysis and abstract algebra.

Content Accuracy rating: 5

I did not encounter any inaccuracies or errors in the book.

Relevance/Longevity rating: 5

After completing the calculus sequence, math majors take many classes in which writing and communication of ideas is more central than calculation. They often struggle with this transition and this book, along with the course it is meant for fills an important niche.

Clarity rating: 5

The goals of the text are clear. The exercises should guide students to an understanding of how to approach the study of mathematics. Solutions to the odd numbered exercises are generally clear and complete.

Consistency rating: 5

The text is consistent - I found no discrepancies while reading.

Modularity rating: 5

Most sections are short enough to digest in one sitting and are clearly written. Some of the material will probably be challenging to some students, but reading mathematics is a process that often requires very slow reading. Learning this process is important for students.

Page x of the introduction contains dependency chart that helps an instructor tailor their course to the needs of their curriculum.

Organization/Structure/Flow rating: 5

Topics are organized in a logical and standard sequence for the subject.

Interface rating: 4

The book is available in two forms, pdf and hard copy.

The pdf is screen enabled so that you can navigate to topics from the table of contents or index. A link in the table of contents to the index would be useful. Within the text, there are links to referenced items (e. g., if a proof uses an earlier numbered theorem, one can click on the theorem number and be sent to the statement). However these links are not apparent unless you try clicking on them. Making the text color of active links a different color would be helpful.

Inexpensive paperback and hard cover versions are available.

Grammatical Errors rating: 5

No issues found.

Cultural Relevance rating: 5

No significant insensitive material was detected in this book. One exercise uses a statement of a biblical nature to make a point of how a statement can be a proposition without having knowledge of its truth. It could bother some, but it could really be applied to any cultural word view. The rest of the book deals abstract ideas that shouldn’t be controversial.

Comments

I'm glad to have encountered this book. I teach a course in mathematical problem solving and plan to use this book as a supplement that will serve as a review of prerequisites to the course and an introduction to a few topics that are included in the course.

Reviewed by David Miller, Professor, West Virginia University on 4/18/19

Comprehensiveness rating: 5 see less

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.

Content Accuracy rating: 5

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.

Relevance/Longevity rating: 5

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).

Clarity rating: 4

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.

Consistency rating: 4

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

Modularity rating: 5

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.

Organization/Structure/Flow rating: 5

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.

Interface rating: 5

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.

Grammatical Errors rating: 5

I have not observed any grammatical errors in the text.

Cultural Relevance rating: 5

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

Comments

I would definitely recommend this textbook to other colleagues.

Reviewed by Michael Barrus, Assistant Professor, University of Rhode Island on 2/1/18

Comprehensiveness rating: 4 see less

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.

Content Accuracy rating: 5

The content is accurate and error-free.

Relevance/Longevity rating: 5

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.

Clarity rating: 5

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.

Consistency rating: 5

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

Modularity rating: 5

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.

Organization/Structure/Flow rating: 5

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.

Interface rating: 4

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.

Grammatical Errors rating: 5

There are no known grammatical errors.

Cultural Relevance rating: 5

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.

Comments

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.

Reviewed by Edwin O'Shea, Associate Professor, James Madison University on 4/11/17

Comprehensiveness rating: 4 see less

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.

Content Accuracy rating: 5

Errors are rare, content is accurate.

Relevance/Longevity rating: 5

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.

Clarity rating: 5

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!).

Consistency rating: 5

The book is consistent in terms of terminology.

Modularity rating: 5

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.

Organization/Structure/Flow rating: 5

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.

Interface rating: 4

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."

Grammatical Errors rating: 5

No grammatical errors.

Cultural Relevance rating: 4

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.

Reviewed by Jess Ellis, Assistant Professor, Colorado State University, Fort Collins on 1/7/16

Comprehensiveness rating: 5 see less

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.

Content Accuracy rating: 5

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

Relevance/Longevity rating: 5

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.

Clarity rating: 5

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.

Consistency rating: 5

Very consistent.

Modularity rating: 5

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.

Organization/Structure/Flow rating: 5

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.

Interface rating: 5

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.

Grammatical Errors rating: 5

No errors.

Cultural Relevance rating: 5

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

Comments

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.

Reviewed by Roberto Munoz-Alicea, Instructor/Academic Support Coordinator, Colorado State University on 1/7/16

Comprehensiveness rating: 5 see less

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.

Content Accuracy rating: 5

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.

Relevance/Longevity rating: 5

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.

Clarity rating: 5

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.

Consistency rating: 5

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.

Modularity rating: 4

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.

Organization/Structure/Flow rating: 4

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 rating: 4

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.

Grammatical Errors rating: 4

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 rating: 5

Cultural relevance is not an issue for this textbook.

Comments

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.

Reviewed by Milos Savic, Assistant Professor, University of Oklahoma on 1/12/15

Comprehensiveness rating: 5 see less

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.

Content Accuracy rating: 5

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.

Relevance/Longevity rating: 5

I love the content of this textbook. Since this topic is relevant for many aspiring mathematicians, the text will live a long life.

Clarity rating: 4

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.

Consistency rating: 5

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

Modularity rating: 4

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.

Organization/Structure/Flow rating: 5

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.

Interface rating: 3

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.

Grammatical Errors rating: 5

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

Cultural Relevance rating: 4

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.

Comments

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.