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How to Prove It: A Structured Approach
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How to Prove It: A Structured Approach

4.21  ·  Rating details ·  550 Ratings  ·  22 Reviews
Geared to preparing students to make the transition from solving problems to proving theorems, this text teaches them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step brea ...more
Paperback, 2nd Edition, 384 pages
Published March 1st 2006 by Cambridge University Press (first published November 25th 1994)
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Simon Vindum
Nov 30, 2015 rated it it was amazing
Shelves: computer-science
This is how math should be thought. It is a very interesting book that explains how mathematical proofs works from the bottom up. In the process of doing that it also teaches discrete math. The learning curve was just right—something that is no easy to achieve. Velleman explains things in a way that is far from being dry yet understandable and precise. I believe everyone who comes in contact with mathematical proofs should read the book. The chapter on induction is especially useful if your fiel ...more
Jul 27, 2010 rated it it was amazing
Shelves: mathematics
С одной стороны, нам все это рассказывали на первом курсе. С другой стороны, никто из преподававших математику на первом курсе не рассказывал это так внятно и с такими хорошими примерами и упражнениями. Так что жаль, что у меня не было этой книги тогда, может быть, я полюбила бы математику нежной любовью %)

И к сожалению, этой книги почему-то нет на русском.

P. S.: еще жаль, что в универе я не знала про сайт %)
Benjamin Schneider
Sep 23, 2017 rated it it was amazing
Working through this book was tremendously rewarding. The book very logically and lucidly explained how proofs work and guides the reader through interesting exercises in logic and useful topics such as set theory and countability. This book is excellent preparation for any rigorous math class that contains proofs (as opposed to just calculations and numerical examples).

This book is very accessible and demands from the student little in the way of prerequisite math knowledge.
Ovais Mohammad
Jun 28, 2016 rated it really liked it
Highly recommended for beginners as it helps tremendously in understanding the mathematical rigour.

Author does not expect much from the reader and begins with very basic concepts and slowly progresses towards quantifiers, then set theory, relation and functions, mathematical induction and finally, infinite sets.

Inside introduction, author gives proof of few theorems in an intuitive way. Later when armed with all the proofing techniques all of those proofs were revisited and reader can clearly se
Dec 30, 2013 rated it it was amazing
"How to Prove It" is a wonderful textbook on the different techniques one can use to prove mathematical theorems using first-year logic. It is very well-written from the point of view of someone with little mathematical knowledge beyond high-school math. As someone who enjoys systematic-thinking, precision and rigour, I truly enjoyed the journey from simple, ordinary proofs to proofs involving different sizes of infinities. And though I didn't quite understand everything, that is because I read ...more
Andi Geng
Dec 30, 2015 rated it it was amazing
I picked this book up because I had zero experience with proofs, and was seriously struggling while trying to learn math. This is a fantastic (and gentle) first exposure to proofs - the book walks you through basic logic, set theory, proof methods, basic number theory, etc.

If you do the exercises, you'll have found that what initially seemed like an arbitrary set of rules have become a set of tools that feel completely natural. Also note that there is no shortage of exercises -- you can do as ma
Andre Harmse
Jul 05, 2012 rated it really liked it
The book delivers what it promises - a structured approach to proofs. It can be a bit challenging, but develops the theory from the ground up and walks the reader through at the beginning. Towards the end, Velleman moves pretty quickly through the material, assuming the reader as absorbed all of the earlier material, which is fine, but it makes for some challenging sections. The progression from sets to relations to functions to cardinality flowed well. There are also many useful interesting exe ...more
Jessica Austin
Dec 14, 2013 rated it it was amazing
Shelves: robotics
Man, I wish I had read this book BEFORE undergrad. In this book, Velleman does three things:
* describes basic concepts in Logic
* gives common proof strategies, with plenty of examples
* dives into more set theory, defining functions, etc

He does all this assuming the reader is NOT a mathematician–in fact, he does an excellent job of explaining a mathematician’s thought process when trying to prove something.

I highly recommend this book if you feel uncomfortable reading and/or writing proofs, since
William Schram
Oct 06, 2014 rated it it was amazing
Shelves: mathematics
This book demonstrates proofs and shows the underlying logical machinery behind them. It focuses especially on the language of mathematical logic. This is a good thing since most of the symbols might as well be from an alien language. It is split into seven chapters with two appendices, a section on suggested further reading, a summary of proof techniques mentioned, and an index. The book also mentions Proof Building Software, but I did not check to see if the link still worked or not.
Chris Ereth
Aug 09, 2015 rated it really liked it
"How to Prove It" is a wonderful textbook on the different techniques one can use to prove mathematical theorems using first-year logic. It is very well-written from the point of view of someone with little mathematical knowledge beyond high-school math. I truly enjoyed the journey from simple, ordinary proofs to proofs involving different sizes of infinities. This text was a great introduction to set theory and mathematical induction.
Reinier Tromp
Sep 26, 2016 rated it really liked it
Great introduction for writing proofs for mathematics. Guides you step by step until you write a perfect proof, providing different methods. Would be a 5-star book if all the exercises would have an answer, offline or online. It matters whether you are right or wrong, right? Also no proof methods that are common in logic and algebra, like Natural Deduction, sequent calculus or axiomatic proof sytems like Hilberts.
May 31, 2011 rated it really liked it
Along with proof methods, this is an excellent explanation of and introduction to symbolic logic. I'm not mathematician, but I am interested in the subject and this book was a key addition to my mathematics library.

The only downside is that, like other non-text books, there are only selected answers to the many exercises throughout the book.
Vilém Zouhar
Dec 07, 2016 rated it really liked it
Intense, yet very educational. If you are familiar with the basics of propositional logic, feel free to skip the first chapter. The excercises provided were sometimes a little out of topic and could easily drown a casual reader, but overall a great book.
Kevin Montes
May 22, 2015 rated it it was amazing
Shelves: math-science
I used this book for an introductory class I took on logic and set theory, and I really enjoyed using it. Easy to understand, a smooth read, and plenty of problems/examples to work through and gauge understanding
Jan 11, 2008 rated it it was amazing
This book should have been read by everyone who took calculus, before they took it. Mathematical induction has been improperly given a sharp learning curve by crappy teachers at my school. For myself and I'm sure many others this book amounts to a course missing from the math curriculum.
Oct 12, 2012 rated it liked it
Great book as Mathematical Thinking books go. Comprehensive and readable. Plenty of examples. doesn't do what many books do, which is to sacrifice accessibility to achieve mathematical elegance.
Dec 11, 2010 rated it really liked it
Awesome; I used it in his class.
Robb Seaton
Sep 28, 2013 rated it it was ok
Exercises are boring, examples are boring, no thrill of discovery.
Vincent Barr
Dec 27, 2013 marked it as to-read
Shelves: pragmatic
Witty, sharp, and helpful for detecting the bologna in your own decision making and - of course - in others'.
Ihab McShea
Oct 14, 2016 rated it really liked it
Heh, that was kind of intense.
May 14, 2011 rated it liked it
Shelves: mathematics
Book seems Ok but I have not found anything new in it
Shaun Zhang
Aug 20, 2016 rated it really liked it
Shelves: mathematics
This is a very great introduction to logic and method of proof. The author exemplifies each method by several interesting and classical problems. It suits the best for the beginners in logic.
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