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How to Prove It: A Structured Approach (Englisch) Taschenbuch – 27. April 2006


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Produktinformation

  • Taschenbuch: 398 Seiten
  • Verlag: Cambridge University Press; Auflage: 2 (27. April 2006)
  • Sprache: Englisch
  • ISBN-10: 0521675995
  • ISBN-13: 978-0521675994
  • Größe und/oder Gewicht: 15,2 x 2,2 x 22,8 cm
  • Durchschnittliche Kundenbewertung: 5.0 von 5 Sternen  Alle Rezensionen anzeigen (1 Kundenrezension)
  • Amazon Bestseller-Rang: Nr. 56.654 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
  • Komplettes Inhaltsverzeichnis ansehen

Produktbeschreibungen

Pressestimmen

'The book begins with the basic concepts of logic and theory ... These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. No background standard in high scholl mathematics is assumed.' L'enseignement mathematique

Über das Produkt

Dan Velleman's lively text prepares students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. This new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software.

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Einleitungssatz
As we saw in the introduction, proofs play a central role in mathematics, and deductive reasoning is the foundation on which proofs are based. Lesen Sie die erste Seite
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15 von 15 Kunden fanden die folgende Rezension hilfreich Von MaM am 24. Juni 2008
Format: Taschenbuch
Die Angelsachsen entlarven mit ihrem Pragmatismus immer wieder unsere Mythen über Geistesleistungen. Es gibt kein deutschsprachiges Buch, das sich annähernd so mit der Methodik des Beweisen auseinandersetzt, wie in diesem Buch. Beutelspachers "Das ist o.B.d.A trivial" macht zwar den Versuch zu erklären, wie ein Beweis korrekt formuliert sein sollte, liefert jedoch keine Informationen darüber, wie er funktioniert. Velleman lehrt sowohl wie man Beweise liest, als auch wie man sie schreibt. Dabei geht es ihm mehr um die Struktur als um die Form.
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Die hilfreichsten Kundenrezensionen auf Amazon.com (beta)

Amazon.com: 63 Rezensionen
71 von 72 Kunden fanden die folgende Rezension hilfreich
This Book Taught Me How to "Get" Math... Please Read On.. 17. Juli 2012
Von Baze - Veröffentlicht auf Amazon.com
Format: Taschenbuch Verifizierter Kauf
Before buying this book, I struggled in math. I excelled at "calculating" stuff by simply plugging in numbers into some sort of equation our high school teachers would spoil us with, but when I got to college, I had to start thinking abstractly- and it bothered me a lot, because I had no idea how to test or prove the logic of some statement. I was doing very poorly in linear algebra and desperately needed help- lo and behold, my professors weren't helpful (at all). Someone recommended this proof writing book to me, and I am VERY grateful for that referral.

The book takes the average student (it's shocking with how little math background one needs) and introduces him to basic boolean logic. You know, material like "If A is true, and B is false, then A implies B is false." In a discrete mathematics course, one would call this "truth tables." From there, the author takes the reader into set theory, basic proofs, group theory, etc- and into more advanced topics, like the Cantor-Schroeder-Bernstein theorem, countability, etc. So what makes this book stand out?

(1) Readability. Many math professors stop just short of taking pride in how confusing, abstract, or daunting their lectures can be. Velleman, however, goes the extra mile in the text to see that the reader UNDERSTANDS the logical buildup and concepts of mathematical proofs. Sure, set theory can be confusing- but after reading several other texts in discrete math, including "Discrete Math and its Applications" by Kenneth Rosen (if you're reading this, no offense) I've found that Velleman by far writes the most comprehensive and cohesive explanations for understanding set theory. Making the material accessible is the mark of a real "teacher," and if you read through this book yourself, I believe you'd agree that Velleman is a pretty legit teacher.

(2) Examples. There are plenty- plenty that Velleman works out himself. Reading the examples alone- and actually taking the time to understand them- is a task that's up to the reader, obviously, but they do show results almost immediately in understanding discrete math.

(3) Problems (exercises). There's never a shortage of exercises, I found, as I tried to work through the problem set. There are plenty. Fortunately, there are some answers in the back, but just enough so that you can verify to see if you're understanding the material, and not enough so that you find yourself copying every answer in the back (even the best students get tempted to do that). Velleman gives the proper amount of answers in the back and a ton of exercises to do. If you complete them all properly, you'd be far ahead of the curve amongst math majors.

I know my review may have been too wordy, or too optimistic. However, my feelings are very honest and not exaggerated: this book is written so one can learn discrete mathematics, and really helps the reader understand what higher math is all about- and how mathematicians think, write, and communicate. This book deserves an A+, and I've only given that score out to a handful of books.
28 von 30 Kunden fanden die folgende Rezension hilfreich
Buy it early in your math major and keep it throughout 1. April 2011
Von Charles Ashbacher - Veröffentlicht auf Amazon.com
Format: Taschenbuch
All math teachers at the college level are familiar with students hitting the "struggling with proofs" wall. Students take calculus and do fairly well using the algorithms to differentiate and integrate functions and this continues into the first part of linear algebra. However, when it is time to understand and execute the proofs they experience a great deal of difficulty that many simply cannot overcome.
This book is designed to present a set of techniques used in mathematical proofs and that aspect is well done. Yet, this book is also just as valuable for the thorough treatment of many of the foundational structures of mathematics. Those topics are:

*) The logic of propositions and predicates
*) Set theory
*) Relations and functions
*) Mathematical induction and recursion
*) Infinite sets

The combination of a thorough introduction to these topics as well as demonstrating proof techniques applied to these objects is an excellent way to learn about them, so this book would be a valuable text in the foundations of mathematics.
The more complex or difficult proofs are also presented in a very stepwise deconstruction, begun using a technique called scratch wok, where even the most insignificant details are included. Once the preliminary scratch work is completed, the formal proof is given. While experienced readers will find this tedious, beginners will find the clarity a relief. A large number of exercises are given at the ends of sections and chapters and solutions to many are included in an appendix.
If you have a course in the foundations of mathematics for the early math major, this is a book that would be an excellent text. It would also be valuable as a supplemental reference text for all students taking a math course where understanding of proofs is required. Think of it as a boost over the wall.
28 von 30 Kunden fanden die folgende Rezension hilfreich
Completely changed my view of proofs. 21. März 2009
Von L. Burton - Veröffentlicht auf Amazon.com
Format: Taschenbuch
Now I understand how proofs are being constructed. I can read and write them the right way! After reading this book I went back to my Calculus textbook and started looking at the proofs. I was amazed at how differently I perceived them. I actually enjoyed reading them and understood why they were written that way.

A little info about the book. Basically, it teaches you the same material that you learn in a Discrete Mathematics course - Propositional logic, Sets and Proofs, Relations, Functions, and Mathematical Induction. However, it looks at those subjects from a completely different perspective. There's absolutely no practical information - all you do is prove stuff.

I strongly advise to learn Discrete Math before reading this book, because getting straight to the proofs of the material, that you just have learned and have no previous experience with, can get very tough.

The first two chapters were a bit boring and too easy - but only because I have already learned that stuff. Chapter 3 is where you start to do your own proofs and is where it gets fun.

The exercises are not hard, and shouldn't present any trouble for the reader. However, I did find the exercises in the last 3 chapters to be more challenging. There were some problems on which I was simply staring for an hour, literally, trying to figure out the way to prove it. The theorem made sense to me, but I couldn't find a way to put into strict mathematical proof! But let me tell you, there's nothing like getting a "Eureka!" moment and figuring out the answer all by yourself. I have just spent 1.5 hours doing 1 problem, and after getting the answer I've felt like I have accomplished something.

Get this book, NOW!
25 von 30 Kunden fanden die folgende Rezension hilfreich
2nd Edition 15. April 2012
Von chaos101 - Veröffentlicht auf Amazon.com
Format: Taschenbuch Verifizierter Kauf
I would've rated this a 4 star or even a 5 if it had more than a few selected exercises answered in the back. Relatively easy to understand book but as far as self studying ... meh! Kind of an annoyance not having feedback that you REALLY understand the content. The kind of feedback you get from doing exercises and verifying your answers are right WITH AN ANSWERKEY! A FULL ANSWERKEY! Otherwise I'll just have to assume I'm right in all my answers and I know that can't be. Upside to this book (there are many) is that you will think you're the smartest person alive. But seriously if you don't want to passively learn math proofs I would go for a book that at least gives answers somewhere to assure you actually understand what you're reading. I say "passively learn" because while it does have the right amount of exercises, you could be getting them all wrong.
8 von 8 Kunden fanden die folgende Rezension hilfreich
Solutions to: Does it work on Kindle? & Are there Solutions to the Exercises? 4. Juli 2014
Von DrTrips - Veröffentlicht auf Amazon.com
Format: Taschenbuch
My goal for this review is to make it as helpful as possible to someone considering to buy this Book !
-I will not go over most of the summaries of the text provided by my fellow reviewers but will provide two important clarifications.
1st:
I read in another review that that the Kindle version of the text interprets some logical operators or other notation incorrectly causing confusion with what the text or exercises are referring to....
This Statement is FALSE. I have purchased the kindle version and did not find one inconsistency and all notations are indeed accurate.
2nd:
I have found that a major complaint about this text is that it does not provide enough solutions to its exercises for one to verify whether they have actually learned the material or not.
This Statement is TRUE. On average, out of 7 question in each section, only 2 solutions are given in the back of the text.

HOWEVER!!! There is another way to circumvent this problem. The Department of Mathematics of the University of California, Santa Barbara has been so kind as to post the Solutions to the unlisted problems on their website.
Please visit this site to view them:
http://www.math.ucsb.edu/~dai/813wang.html

Now with both of these clarifications in place, and after going through a couple of other Mathematical Proof books, and I personally prefer this one. It is direct, and covers the basics needed for understanding and doing proofs.
One must understand that doing proofs is a skill a Mathematician gains through vast experience, practice and long hours of thought. Finding a book that breaks down a universal method of proofing in the same simple way an Algebra text shows how to use a formula will not be possible. Mainly because there is no universal method of proofing. All one can then hope to do in order to understand higher level of mathematics, or what I would describe better as the underlying foundation of mathematics, is to understand the basic language of proofs, their structure, and their organization enough for the reader to go forth and gain their own experience.
And this book does precisely that
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