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Proofs and Fundamentals: A First Course in Abstract Mathematics [Hardcover]

by Ethan D. Bloch
3.5 out of 5 stars  See all reviews (11 customer reviews)

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There is a newer edition of this item:
Proofs and Fundamentals: A First Course in Abstract Mathematics (Undergraduate Texts in Mathematics) Proofs and Fundamentals: A First Course in Abstract Mathematics (Undergraduate Texts in Mathematics) 3.5 out of 5 stars (11)
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Book Description

June 1, 2000 0817641114 978-0817641115 1

The aim of this book is to help students write mathematics better. Throughout it are large exercise sets well-integrated with the text and varying appropriately from easy to hard. Basic issues are treated, and attention is given to small issues like not placing a mathematical symbol directly after a punctuation mark. And it provides many examples of what students should think and what they should write and how these two are often not the same.


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Editorial Reviews

Review

". . . Proofs and Fundamentals has many strengths. One notable, strength, is its excellent organization. The book begins with a three-part preface, which makes its aims very clear. There are large exercise sets throughout the book . . . Exercises are well integrated with the text and vary appropriately from easy to hard . . . Topics in Part III are quite varied, mostly independent from each other, and truly dependent on Parts I and II. At the end of the book there are useful hints to selected exercises.

Perhaps the book’s greatest strength is the author’s zeal and skill for helping students write mathematics better. Careful guidance is given throughout the book. Basic issues like not abusing equal signs are treated explicitly. Attention is given to even relatively small issues, like not placing a mathematical symbol directly after a punctuation mark. Throughout the book, theorems are often followed first by informative ‘scratch work’ and only then by proofs. Thus students can see many examples of what they should think, what they should write, and how these are usually not the same."

–MAA Online

"This is a well-written book, based on very sound pedagogical ideas. It would be an excellent choice as a textbook for a 'transition' course." ---Zentralblatt Math

From the Back Cover

This textbook is designed to introduce undergraduates to the writing of rigorous mathematical proofs, and to fundamental mathematical ideas such as sets, functions, relations, and cardinality. The book serves as a bridge between computational courses such as calculus and more theoretical courses such as linear algebra, abstract algebra, and real analysis.

This second edition has been significantly enhanced, while maintaining the balance of topics and careful writing of the previous edition. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material such as sets, functions and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences, and suggests avenues for independent student explorations.

A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised.

Reviews of the first edition:

This is a well-written book, based on very sound pedagogical ideas. It would be an excellent choice as a textbook for a 'transition' course.
—Zentralblatt Math

'Proofs and Fundamentals' has many strengths. One notable strength is its excellent organization... There are large exercise sets throughout the book... the exercises are well integrated with the text and vary appropriately from easy to hard... Perhaps the book’s greatest strength is the author’s zeal and skill for helping students write mathematics better.
—MAA Online

--This text refers to an alternate Hardcover edition.

Product Details

  • Hardcover: 424 pages
  • Publisher: Birkhäuser; 1 edition (June 1, 2000)
  • Language: English
  • ISBN-10: 0817641114
  • ISBN-13: 978-0817641115
  • Product Dimensions: 9.5 x 6.5 x 1 inches
  • Shipping Weight: 1.7 pounds (View shipping rates and policies)
  • Average Customer Review: 3.5 out of 5 stars  See all reviews (11 customer reviews)
  • Amazon Best Sellers Rank: #849,472 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
38 of 41 people found the following review helpful
Format:Hardcover
"Proofs and Fundamentals" by Ethan Bloch is undoubtly one of the few books I'd say would be required when beginning (and a light refrence as you progress) the undergraduate mathematics curriculum in most US schools.

There is some bit of controversay about "Foundations" courses in general. It use to be the case that there was no 'bridge' course that linked the more applied Calculus classes (something like your average 'multi-variable calculus' withi something more theory based such as Abtract Algebra)

The basics of proof writing were usually taught in your first theory course. In some institutions (I hear Cornell as one of them) still teach your basic proof writing in their first theory course. However, much of the large research state flagship schools in the country and even some of the older ivies such as Princeton (which I hear offers 2 - 3 foundations courses depending on your mathematical taste) now have come around to the concept of a bridge course in mathematics.

To be sure, the refinement of the curriculum to include some kind of bridge course in most departments in the US has led (in my opinion) to the ability for more students to study mathematics.

With that in mind, this book is wonderful for that purpose for the following reasons:

1. Its very readable
2. It gives an introduction to propositional logic and naive
set theory.
3. It covers numerous fundemental topics that is the start to
some very interesting mathematics.

To comment on the first point briefly, the author does utilize the standard definiton-theorem-proof-excercise template, however he addds alot of motivating exposition in between. What is motivating exposition?
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33 of 36 people found the following review helpful
5.0 out of 5 stars an excellent introduction to formal mathematics May 4, 2001
By A Customer
Format:Hardcover
This book is an excellent and thorough intorduction to the world of formal mathematics. Typically, a new mathematics student finds himself or herself picking up random mathematical concepts and techniques of proof along the way, without ever having the chance to sit down and go through the fundamentals of formal mathematics and proof. This book covers a little of everything, and has a thorough introduction to sets, functions, inverses, equivalence and order relations. There are also sections on introductory number theory, algebra, combinatorics, logic, and much more. It provides an excellent overview for the student who will be using these tools on a daily basis, for the layman who is interested in finding out what mathematics is really about, or for the seasoned mathematician who needs a good general reference book. There are also extensive and thorough sections on the construction and writing of mathematical proofs -- somthing on which many new and not-so-new mathematicians could use some improvement.
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13 of 14 people found the following review helpful
2.0 out of 5 stars Good on Fundamentals, Not So Good on Proofs April 15, 2008
By MRJ
Format:Hardcover
This book offers brief coverage of an impressive array of "fundamental" topics that are likely to come up in various later courses. The value here is that in most later (upper division) math classes there is a major benefit to knowing even a little of the material from some of the other upper division classes. On the other hand, this book simply will not help you learn how to do proofs. After going about half way through the book and working on the exercises, I realized that I was no better at doing proofs than when I began. The "hints" to the exercises in the back of the book are really no help at all for self study. I usually know the right strategy, what I really need to learn is the correct way to express my ideas as a proof, and this book does not provide very much guidance on how to do that.
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2 of 2 people found the following review helpful
5.0 out of 5 stars Best "proofs" book November 23, 2010
Format:Hardcover|Verified Purchase
I own a few proofs/transitions books. I have worked through Bloch's Proofs and Fundamentals (the first edition) and the book by Daniel Solow. Bloch has written the best and clearest book for self-study of proofs. The back of the book has over 20 useful pages of Hints for Selected Exercises. For those reviewers who think Bloch is too wordy, perhaps he is if you already have an instructor in a class explaining everything to you. But if you are on your own, trying to learn by working through a book, Bloch's explanations are just right.
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1 of 1 people found the following review helpful
4.0 out of 5 stars Great introduction for novices December 3, 2009
Format:Hardcover
I picked this book up one day in the bookstore, started reading and left with it. I was a math major at one point in college, but I moved on for other reasons. These days most math programs offer courses such as "Introduction to Proofs." This book serves that purpose. Now that I am back in college taking math courses again, I feel well prepared in my proof-oriented classes after going through about half of this book. The author does an excellent job of stressing the importance of proofs through language that is easy to understand. There is also a lot to read in the sense that concepts are explained and discussed in English. The exercises are adequate, although it would be better if more solutions were offered in the book (only select ones are). I did most of the exercises and that is where the real learning took place.

This book would not be useful for someone who has any kind of a background in proofs/fundamentals. It's for those who have only taken computation-oriented classes who want to learn how to do basic proofs. I would recommend finding a copy and opening it up to see if it fits your style. Anyone could learn from the beginning of this text, but it takes some mathematical sophistication to get through more advanced parts. The text seems technically precise (clarity on the difference between range and codomain for instance) and remains reader-friendly throughout. The author also does not hesitate on giving his opinions on how math proofs should be written. I liked the way that he emphasized writing proofs in complete sentences, and in keeping the scratch work and attempts at a proof separate from the proof itself. Also, despite the language, rigor and mathematical symbolism are not sacrificed.
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