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Reviewed by David Miller, Professor, West Virginia University on 4/18/19Comprehensiveness rating: 5This 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.Accuracy rating: 5I 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: 5The 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: 4The 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: 4The textbook is very consistent except on the aspect listed in the previous comment about paragraph proofs.Modularity rating: 5The 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: 5The 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: 5I 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: 5I have not observed any grammatical errors in the text.Cultural Relevance rating: 5I haven't found any text in the book that is culturally insensitive or offensive.CommentsI would definitely recommend this textbook to other colleagues. Reviewed by Edwin O'Shea, Associate Professor, James Madison University on 4/11/17Comprehensiveness rating: 4The 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.
Reviewed by Roberto Munoz-Alicea, Instructor/Academic Support Coordinator, Colorado State University on 1/7/16Comprehensiveness rating: 5This 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.Accuracy rating: 5While 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: 5The 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: 5The 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: 5The 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: 4This 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: 4Even 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: 4Interface 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: 4There 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: 5Cultural relevance is not an issue for this textbook.CommentsThis book is excellent for an introduction to mathematics proofs course.
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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 Jess Ellis, Assistant Professor, Colorado State University, Fort Collins on 1/7/16Comprehensiveness rating: 5I 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.Accuracy rating: 5After two semesters of teaching from it I have not found an error.Relevance/Longevity rating: 5Because 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: 5Very 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: 5Very consistent.Modularity rating: 5The 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: 5Very 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: 5No 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: 5No errors.Cultural Relevance rating: 5Not applicable? So - it is vacuously highly culturally relevant.CommentsI 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 Milos Savic, Assistant Professor, University of Oklahoma on 1/12/15Comprehensiveness rating: 5This 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.Accuracy rating: 5I 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: 5I love the content of this textbook. Since this topic is relevant for many aspiring mathematicians, the text will live a long life.Clarity rating: 4I 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: 5The text is consistent in framework and terminology. I found no discrepancies while reading.Modularity rating: 4There 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: 5I 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: 3The 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: 5This text contains no grammatical errors from what I've observed.Cultural Relevance rating: 4There 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.CommentsAlthough 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.
Reverse Mathematics by John Stillwell Summary' This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis-finding the 'right axioms' to prove fundamental theorems-and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers.
The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets.
Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the 'right axiom' to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics. Book of Proof by Richard H. Hammack SummaryThis book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.
A Logical Introduction to Proof by Daniel W. Cunningham SummaryThe book is intended for students who want to learn how to prove theorems and be better prepared for the rigors required in more advance mathematics. One of the key components in this textbook is the development of a methodology to lay bare the structure underpinning the construction of a proof, much as diagramming a sentence lays bare its grammatical structure. Diagramming a proof is a way of presenting the relationships between the various parts of a proof. A proof diagram provides a tool for showing students how to write correct mathematical proofs. Proofs Without Words by Malcolm Scott MacKenzie,Roger B. Nelsen SummaryProofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it.
While in some proofs without words an equation or two may appear to help guide that process, the emphasis is clearly on providing visual clues to stimulate mathematical thought. The proofs in this collection are arranged by topic into five chapters: Geometry and algebra; Trigonometry, calculus and analytic geometry; Inequalities; Integer sums; and Sequences and series. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics. Discrete Mathematics by Martin Aigner SummaryThe advent of fast computers and the search for efficient algorithms revolutionized combinatorics and brought about the field of discrete mathematics. This book is an introduction to the main ideas and results of discrete mathematics, and with its emphasis on algorithms it should be interesting to mathematicians and computer scientists alike. The book is organized into three parts: enumeration, graphs and algorithms, and algebraic systems.
There are 600 exercises with hints andsolutions to about half of them. The only prerequisites for understanding everything in the book are linear algebra and calculus at the undergraduate level.
Praise for the German edition. This book is a well-written introduction to discrete mathematics and is highly recommended to every student ofmathematics and computer science as well as to teachers of these topics.Konrad Engel for MathSciNet Martin Aigner is a professor of mathematics at the Free University of Berlin.
He received his PhD at the University of Vienna and has held a number of positions in the USA and Germany before moving to Berlin. He is the author of several books on discrete mathematics, graph theory, and the theory of search. The Monthly article Turan's graph theorem earned him a 1995 Lester R. Ford Prize of theMAA for expository writing, and his book Proofs from the BOOK with Gunter M.
Ziegler has been an international success with translations into 12 languages. 99 Variations on a Proof by Philip Ording SummaryAn exploration of mathematical style through 99 different proofs of the same theorem This book offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice of mathematics. Inspired by the experiments of the Paris-based writing group known as the Oulipo—whose members included Raymond Queneau, Italo Calvino, and Marcel Duchamp—Ording explores new ways to examine the aesthetic possibilities of mathematical activity.
99 Variations on a Proof is a mathematical take on Queneau’s Exercises in Style, a collection of 99 retellings of the same story, and it draws unexpected connections to everything from mysticism and technology to architecture and sign language. Through diagrams, found material, and other imagery, Ording illustrates the flexibility and creative potential of mathematics despite its reputation for precision and rigor. Readers will gain not only a bird’s-eye view of the discipline and its major branches but also new insights into its historical, philosophical, and cultural nuances.
Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape. Proofs and Refutations by Imre Lakatos,Lakatos Imre SummaryProofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.
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How to Prove It by Daniel J. Velleman SummaryMany mathematics students have trouble the first time they take a course, such as linear algebra, abstract algebra, introductory analysis, or discrete mathematics, in which they are asked to prove various theorems.
This textbook will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs.
The author shows how complex proofs are built up from these smaller steps, using detailed 'scratchwork' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. Numerous exercises give students the opportunity to construct their own proofs. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.
Formal Proofs in Maths by Chris Lavranos,Labros Batalas,Konstantinos Lamogiannis SummaryThe scope of Formal Proofs in Maths is to teach students between higher school classes and University undergraduate or postgraduate studies, how to write a formal proof with the true meaning of the concept, of simple theorems in Algebra, particulary in identities concerning equalities, equations and inequalities. This is accomplished by writing four different types of proof namely type(A), type(B), type(C) and type(D) for each theorem or exercise.
In TYPE(A) ordinary proofs will be cited in the usual narrative style used by experienced mathematicians. In TYPE(B) a rigorous proof in steps will be introduced to the reader. Each line of that proof will be justified by an appropriate axiom, theorem or definition. In TYPE(C) we will try for a smooth transition from a rigorous proof to a formal proof exposing the way that the laws of logic apply on one or more statements of the proof. In TYPE(D) we will simply write in tabular stepwise form, the results of TYPE(C) mentioning both: 1) Axioms, theorems or definitions. 2) The laws of logic. Hence, finally producing a formal proof according to the definition given in the preface note of the book.
Introduction to Mathematical Proofs by Charles Roberts SummaryShows How to Read & Write Mathematical Proofs Ideal Foundation for More Advanced Mathematics Courses Introduction to Mathematical Proofs: A Transition facilitates a smooth transition from courses designed to develop computational skills and problem solving abilities to courses that emphasize theorem proving. It helps students develop the skills necessary to write clear, correct, and concise proofs. Unlike similar textbooks, this one begins with logic since it is the underlying language of mathematics and the basis of reasoned arguments. The text then discusses deductive mathematical systems and the systems of natural numbers, integers, rational numbers, and real numbers. It also covers elementary topics in set theory, explores various properties of relations and functions, and proves several theorems using induction. The final chapters introduce the concept of cardinalities of sets and the concepts and proofs of real analysis and group theory. In the appendix, the author includes some basic guidelines to follow when writing proofs.
Written in a conversational style, yet maintaining the proper level of mathematical rigor, this accessible book teaches students to reason logically, read proofs critically, and write valid mathematical proofs. It will prepare them to succeed in more advanced mathematics courses, such as abstract algebra and geometry.
Machine Proofs in Geometry by Shang-Ching Chou,Xiao-Shan Gao,Jingzhong Zhang SummaryThis book reports recent major advances in automated reasoning in geometry. The authors have developed a method and implemented a computer program which, for the first time, produces short and readable proofs for hundreds of geometry theorems.The book begins with chapters introducing the method at an elementary level, which are accessible to high school students; latter chapters concentrate on the main theme: the algorithms and computer implementation of the method.This book brings researchers in artificial intelligence, computer science and mathematics to a new research frontier of automated geometry reasoning. In addition, it can be used as a supplementary geometry textbook for students, teachers and geometers.
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By presenting a systematic way of proving geometry theorems, it makes the learning and teaching of geometry easier and may change the way of geometry education.