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

Ethan D. Bloch (Author)

Book Description

ISBN-10: 0817641114 | ISBN-13: 978-0817641115 | Publication Date: April 20, 2000 | Edition: 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.

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.

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Product Details

• Hardcover: 445 pages
• Publisher: Birkhäuser Boston; 1 edition (April 20, 2000)
• Language: English
• ISBN-10: 0817641114
• ISBN-13: 978-0817641115
• Product Dimensions: 9.3 x 6.2 x 1 inches
• Shipping Weight: 1.7 pounds (View shipping rates and policies)
• Average Customer Review: 3.6 out of 5 stars  See all reviews (8 customer reviews)
• Amazon Best Sellers Rank: #533,859 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

36 of 39 people found the following review helpful:

5.0 out of 5 stars The (Required) Undergraduate Handbook for Beginning Mathematics, January 13, 2006

By 

Irrational Expectations "Chris" (Georgia) - See all my reviews

This review is from: Proofs and Fundamentals: A First Course in Abstract Mathematics (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? Well, Bloch writes (throughout the text) why we (the student) are learnign these concepts and how they fit into the various structures in mathematics.

He dosn't only talk about how concepts/theorems are relevent to theory, but mentions why a certain style maybe relevent to communication. One of Bloch's emphasis is the "communication" aspect of mathematics. That is, it is good to get good notation and adequate writing habits in the beginning (the later is something I have sort slagged on, as I'm more of a concise symbol man myself).

On reason 2, he gives a very good introduction to propositional logic. He even lists most of the rules of inferences early on. This section helped me tremendously. I was always curious on what was the final justification of many mathematical statements (which of course in the naive sense is just proof table verification). Learning this fact allowed me to enjoy mathematics more as it wasn't so much hand waving as a system of verifable statements (which makes math unique of all the subjects you could study).

After this chapter Bloch has a section for "strategy" in proofs. This section is more useful for the basic courses you will take immediatly after you take your foundations, but I honestly don't remember getting much of an impression while I was reading it. Stick to learning the naive foundational material and to doing lots of problems and you'll learn your own strategies. If you want a more concrete book on problem solving a good introduction to that is the "The Art and Craft to Problem Solving."

Of course the next section is naive set theory. Bloch does alot of work for you (and rightly so) in proving most of the results he mentions. Thus it makes this section very smooth reading. In fact, I do not think I did more then a handful of excercises and yet was able to grasp and use the material taught in the book (for most mathematics textbooks this is not possible).

Finally, the author provides the basics of topics that can lead to interesting studies into various differnt mathematics. A chapter of counting principles is included (Combinatorics), Number Systems (Real Analysis), Groups and Lattices (Algebra) and some small filler chapters in between like a chapter on binary operations, Recursion, Cardinality, Induction, and Fuzzy logic to name some.

I got a very good apprecaition for the Foundations from this book and I blame it for giving me my enthusiasm for Mathematical Logic. There are others which are better or worst on some topics, for instance, those who are interested mostly in Analysis may not find this book appealing (A good pure Analysis undergraduate handbook is 'Fundamentals of Analysis') but on balance, this book is good (I'd say excellent). I'd think the best way to utilize it would be to read it the summer before you start rigorious mathematics. Try some of the excercises and then attempt to correct your excercises (determine if they are right or wrong yourself). Then when you take your foundations course (or your first theory course), consult your instructor on your questions. I feel if one sticks to this regiment, they will be well on their way to a succesful curriculum in mathematics.

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

This review is from: Proofs and Fundamentals: A First Course in Abstract Mathematics (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.

8 of 8 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 (USA) - See all my reviews

This review is from: Proofs and Fundamentals: A First Course in Abstract Mathematics (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.