Discrete Mathematics [Paperback]

Laszlo Lovasz (Author), Jozsef Pelikan (Author), Katalin L. Vesztergombi (Author)

Book Description

ISBN-10: 0387955852 | ISBN-13: 978-0387955858 | Publication Date: January 27, 2003 | Edition: 1

Aimed at undergraduate mathematics and computer science students, this book is an excellent introduction to a lot of problems of discrete mathematics. It discusses a number of selected results and methods, mostly from areas of combinatorics and graph theory, and it uses proofs and problem solving to help students understand the solutions to problems. Numerous examples, figures, and exercises are spread throughout the book.

Editorial Reviews

Review

From the reviews: "The goal of this book is to use the introduction to discrete mathematics … . Consequently, the authors … take a lot of time to explain proof techniques and to motivate definitions and style. The language is very informal and easy to read. The level is always introductory which makes it possible to give a taste of a wide range of topics … . There are a lot of exercises … which makes it perfectly suitable for self-study." (T. Eisenkölbl, Monatshefte für Mathematik, Vol. 144 (2), 2005) "The book is aimed at undergraduate mathematics and computer science students interested in developing a feeling for what mathematics is all about, where mathematics can be helpful, and what kinds of questions mathematicians work on. The authors discuss a number of selected results and methods of discrete mathematics … . Wherever possible, the authors use proofs and problem solving to help students understand the solutions to problems. In addition, there are numerous examples, figures, and exercises spread throughout the book." (Zentralblatt für Didaktik der Mathematik, January, 2004) "The title of this book is quite apposite … . The text is, in fact, based on introductory courses in discrete mathematics … . the emphasis throughout the book is on finding efficient and imaginative ways to tackle problems and to develop general results. … I would see it as a valuable resource of enrichment activities for students … . is eminently suited for self-study (there are plenty of exercises and solutions) and can be warmly recommended for the school library." (Gerry Leversha, The Mathematical Gazette, Vol. 88 (512), 2004) "This book is an excellent introduction to a lot of problems of discrete mathematics. … The authors discuss a number of selected results and methods, mostly from the areas of combinatorics and graph theory … . This book is appealed to a broad range of readers, including students and post-graduate students, teachers of mathematics, mathematical amateurs. The authors use proofs and problem solving to help students understand the solutions to problems. In addition, there are numerous examples, figures and exercises spread throughout the book." (M.I Yadrenko, Zentralblatt MATH, Issue 1017, 2003) "This book is aimed at undergraduate mathematics and computer science students interested in developing a feeling for what mathematics is all about, where mathematics can be helpful, and what kinds of questions mathematicians work on. The authors discuss a number of selected results and methods of discrete mathematics … . Wherever possible, the authors use proofs and problem solving to help students understand the solutions to problems. In addition, there are numerous examples, figures, and exercises spread throughout the book." (L’ Enseignement Mathematique, Vol. 49 (1-2), 2003) "The aim of this book is NOT to cover discrete mathematics in depth. Rather, it discusses a number of selected results and methods … . The authors develop most topics to the extent that they can describe the discrete mathematics behind an important application of mathematics … . Another feature that is not covered in other discrete mathematics books is the use of ESTIMATES … . There are questions posed in the text and problems at the end of each chapter with solutions … ." (The Bulletin of Mathematics Books, Issue 43, February, 2003)

From the Back Cover

 Discrete mathematics is quickly becoming one of the most important areas of mathematical research, with applications to cryptography, linear programming, coding theory and the theory of computing. This book is aimed at undergraduate mathematics and computer science students interested in developing a feeling for what mathematics is all about, where mathematics can be helpful, and what kinds of questions mathematicians work on. The authors discuss a number of selected results and methods of discrete mathematics, mostly from the areas of combinatorics and graph theory, with a little number theory, probability, and combinatorial geometry. Wherever possible, the authors use proofs and problem solving to help students understand the solutions to problems. In addition, there are numerous examples, figures and exercises spread throughout the book. László Lovász is a Senior Researcher in the Theory Group at Microsoft Corporation. He is a recipient of the 1999 Wolf Prize and the Gödel Prize for the top paper in Computer Science. József Pelikán is Professor of Mathematics in the Department of Algebra and Number Theory at Eötvös Loránd University, Hungary. In 2002, he was elected Chairman of the Advisory Board of the International Mathematical Olympiad. Katalin Vesztergombi is Senior Lecturer in the Department of Mathematics at the University of Washington.

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Product Details

• Paperback: 296 pages
• Publisher: Springer; 1 edition (January 27, 2003)
• Language: English
• ISBN-10: 0387955852
• ISBN-13: 978-0387955858
• Product Dimensions: 9.1 x 6 x 0.6 inches
• Shipping Weight: 15.2 ounces (View shipping rates and policies)
• Average Customer Review: 3.8 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #316,200 in Books (See Top 100 in Books)

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21 of 22 people found the following review helpful:

5.0 out of 5 stars unique in approach, December 4, 2006

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Hernadvolgyi Istvan (Hungary - Canada) - See all my reviews
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This review is from: Discrete Mathematics (Paperback)

I am really surprised at my fellow reviewer's statements indicating that you need to be a genius to understand this book. In fact, it is really the opposite; the authors took an effort to make the material approachable to the mathematically minded and provide motivating context for each example. While at the authors, you should note that these people are some of the most well known researchers in this area and Dr. Lovasz is also an exceptional lecturer. I believe all possess Erdos number 1 :) It is surely not a textbook, in the sense of Rosen's "Discrete Mathematics and its Applications" nor it strives for completeness like Reinhard Diestel's "Graph Theory". Instead it is a selection of topics that give a good introduction into discrete mathematics with carefully selected insightful problems with solution hints! So, yes, I think it is great for self study and especially for those (as the introduction suggests) who have had a more analysis-biased introduction into Mathematics. Instead of being a collection of theorems and proofs, the problems in this book build on the absolute necessary basics (often just high-school math) and, yes, skip unnecessary notation and pseudo-rigor. I should also note that I am basing this review on the Hungarian edition, which also reads well but I have not actually seen the original English text.

1 of 1 people found the following review helpful:

4.0 out of 5 stars A Skim Along the Surface of Discrete Math, December 22, 2010

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J. Bodah - See all my reviews
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This review is from: Discrete Mathematics (Paperback)

This books is a decent introduction to discrete mathematics. Lovasz does a good job of making material easier by putting it into words. This unfortunately comes at a cost though. For example, in the first few chapters about combinatorics Lovasz does a good job of distinguishing permutations from combinations. However, when he tries to present proofs in every day language the lack of mathematical preciseness can get really confusing. This unfortunately only gets worse as more topics get introduced. The section about fast modular exponentiation is very dense and requires careful reading to follow the math. I feel like these topics could've been presented better if Lovasz simply wrote out the equations and the manipulations. The sections on graph theory and convex geometry go a bit too fast. They start off quite easy and then ramp up rapidly at the end of the chapter which leaves the reader with more questions than answers. The section on RSA was surprisingly good and really brought Fermat's theorem to life, but I do wish that this was done nine chapters earlier.

So, I've complained a lot, and you may wonder why I've given the book four stars. The reason is that the book fulfilled it's purpose very well; it gave me a brief introduction to the many fields of discrete math without totally burying me. The tone and style was easy enough for me to read in my leisure time while still introducing to me some solid mathematical concepts. Most of the basic theorems were very clear (though the more advanced ones were typically presented poorly like I said). Exercises were generally easy and reinforced the topics in the chapter. One of my favorite things about the book was the number of open problems Lovasz explained. More authors should present these in order to stimulate the reader. Overall there's a lot of good stuff in Lovasz's book if you're just curious about what discrete math is about. Unfortunately, if you want a deeper understanding of a topic, you won't really be able to get it here, but that was never the intention. If you're looking for a cheap, brief, casual introduction to discrete math, I would highly recommend this book. It's flaws are glaring, but, considering that the author merely wants to expose the reader to the material at an unintimidating level, it does a good job at dipping the reader's feet into modern problems.

Note: I too had this book assigned as the main text for my discrete course. Even though Lovasz is usually easy to understand, the lack of depth of this book makes it a poor choice for a main text. This book would work much better as a supporting text to teach concepts as opposed to teaching outlying examples and rigorous proofs.

28 of 51 people found the following review helpful:

2.0 out of 5 stars If You're Brilliant and Don't Mind a Lack of Rigor, Try It., December 21, 2004

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David A. Lessnau (USA) - See all my reviews
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This review is from: Discrete Mathematics (Paperback)

Ouch. I definitely made a mistake trying this book. From what I can see, the only set of people who might find this book useful would be genius non-Math-types. My handy-dandy way of explaining this is by mapping the universe of all possible readers onto a set of x-y axes. Let the x axis run from "non-Math-types" up through "Math-types." Let they y axis go from "non-geniuses" up through "geniuses:"

- Quadrant I: if you're a genius Math-type, you'll be aghast at the lack of rigor in the proofs and at all the steps missing from the few proofs given. But, you might be able to work through the material on your own. My guess, though, is that you'll throw the book across the room in disgust, instead.

- Quadrant II: if you're a genius non-Math-type, you might find the lack of rigor in the proofs tolerable. Plus, you, like the Quadrant Is, might be able to work through the material enough so that you can follow the author's explanations. If you're smart enough and have enough familiarity with the material before reading this book, you might find its terseness acceptable.

- Quadrant III: if you're a non-genius non-Math-type (i.e., "normal"), you'll be lost. You (we) won't understand what the point of this is, won't be able to work thru all the missing steps in the few proofs given, and will rant and rave when the authors "prove" one thing, give one example, and then ask the reader to prove the several follow-up theorems as an exercise. With, essentially, one example to work from in a subset of each topic, you'll never be able to work through the few questions with answers (which answers are short to the point of worthlessness) and will throw up your hands in dismay when attempting to solve the non-answered end-of-chapter questions.

- Quadrant IV: if you're a non-genius Math-type, you will join the Quadrant Is in horror at the lack of rigor, but, like the Quadrant IIIs, will be horribly frustrated since you won't be able to force your way through the material on your own. A definite lose-lose situation, here.

The really odd thing with this book is that, in the universities, Discrete Math (the subject) is usually a required course for Computer Science majors and is a PRE-CALCULUS course. There's absolutely no way that any such student at that point in his degree could get through this book. I received my B.S. in Mathematics from the University of Michigan well over 20 years ago (so I've forgotten most everything, plus I was a Quadrant IV type but figured it out much later and migrated to Quadrant III), but there are entire chapters in this book where I only understand words like "the" and "and." The authors assume an extremely in-depth degree of mathematical knowledge on the part of the reader. Also, there's nothing to tie the subject matter back to anything a CS person (or any non-math-type) could use in his degree, profession, or life. So, once again, this book is very poorly matched against any intended audience.

I'd also like to point out that I doubt this book would ever be chosen by any academic institution for teaching. It just doesn't follow the established norms on how to teach. Specifically, it should: 1) tell the readers what it's going to say, 2) tell them, and 3) tell them what it just told them. Each chapter and section in this book just starts out talking about something. You really have no idea where the authors are going until they actually get there. What the book really, REALLY needs is for the authors to state exactly what they're going to do in a section and chapter, do it, and then summarize it. Ditto for the book as a whole. The preface needs some kind of overall game plan so the people reading the book know where they're going.

Since the publisher hasn't provided the information on Amazon, I've put a scanned copy of the book's Table of Contents in the "User Images" area at the top of the page.

I rate this book 2 stars out of 5.