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Designed for students preparing to engage in their first struggles to understand and write proofs and to read mathematics independently, this is well suited as a supplementary text in courses on introductory real analysis, advanced calculus, abstract algebra, or topology. The book teaches in detail how to construct examples and non-examples to help understand a new theorem or definition; it shows how to discover the outline of a proof in the form of the theorem and how logical structures determine the forms that proofs may take. Throughout, the text asks the reader to pause and work on an example or a problem before continuing, and encourages the student to engage the topic at hand and to learn from failed attempts at solving problems. The book may also be used as the main text for a "transitions" course bridging the gap between calculus and higher mathematics. The whole concludes with a set of "Laboratories" in which students can practice the skills learned in the earlier chapters on set theory and function theory.
- Sales Rank: #2460178 in eBooks
- Published on: 1996-11-30
- Released on: 1996-11-30
- Format: Kindle eBook
Review
Though I am neither a student nor an instructor of mathematics (although I use mathematics in my teaching and apply it in my work), I found this book extremely useful and fun to read. The language is colorful, the phrasing is simple, and the facts are not intimidating. At times, I felt that I was reading a good novel, whose characters were definitions, examples, rules, proofs, and such, and that I had a magic stick to make them behave according to my will. -- T. Rus, Computing Reviews, May 1997
From the Back Cover
This text prepares undergraduate mathematics students to meet two challenges in the study of mathematics, namely, to read mathematics independently and to understand and write proofs. The book begins by teaching how to read mathematics actively, constructing examples, extreme cases, and non-examples to aid in understanding an unfamiliar theorem or definition (a technique familiar to any mathematician, but rarely taught); it provides practice by indicating explicitly where work with pencil and paper must interrupt reading. The book then turns to proofs, showing in detail how to discover the structure of a potential proof from the form of the theorem (especially the conclusion). It shows the logical structure behind proof farms (especially quantifier arguments), and analyzes, thoroughly, the often sketchy coding of these forms in proofs as they are ordinarily written. The common introductory material (such as sets and functions) is used for the numerous exercises, and the book concludes with a set of "Laboratories" on these topics in which the student can practice the skills learned in the earlier chapters. Intended for use as a supplementary text in courses on introductory real analysis, advanced calculus, abstract algebra, or topology, the book may also be used as the main text for a "transitions" course bridging the gap between calculus and higher mathematics.
Most helpful customer reviews
21 of 21 people found the following review helpful.
Good book on proofs
By A Customer
I really appreciate "An accompaniement to higher mathematics", because it presents a great amount of stuff concerning the technique of proof. The first chapter present why examples are so importants by showing how to test definitions, how to construct good and bad examples and how to test the validity of a theorem by the mean of extreme example. The material in this chapter is basic and easy to follow. The second chapter is about the infirmal language and some technics of proof. First, it presents the logic behind the proofs. Also, this chapter presents technics such as induction, proof by case, differents forms of proof based on implication. The third chapter is about the use of quantifier in the proof. It shows how to use and when to use quantifier. Also how to find the structure of a proof. Why I find this book interresting ? First the book is full of exercises of different kind (set, function, analysis, ...and the forth chapter contain laboratories that give you again plenty of exercises), It is written clearly, the author give a lot of advice about proofs, I find the book very suitable to undergraduted, I find the style of writting of the book very motivating. This book is definitely a good one but it is not perfect. I found some of the exercises too easy (about 30 % of the exercises), I don't think that this book is suitable for graduate student but it may help in the way you work proofs and problems. Also this book give me some help in analysis course. I recommend this book for anyone who want to learn the basic and more about proofs.
3 of 3 people found the following review helpful.
Great Introduction
By Buck Hef
This is simply a great text for introducing undergraduate students to the basics of upper-level mathematics. Stresses the importance of examples and definitions in proof discovery. While probably inappropriate for graduate students, it makes a great primary text for any first course in proofs at the undergraduate level and is written primarily to students in this situation. Overall, the book seems to be extremely appreciated by students transitioning from calculus to upper-level mathematics.
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