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How to Prove It: A Structured Approach
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 breakdown of the most important techniques used in constructing proofs. To help students construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. Previous Edition Hb (1994) 0-521-44116-1 Previous Edition Pb (1994) 0-521-44663-5
384 pages, Paperback
First published November 25, 1994
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Displaying 1 - 30 of 54 reviews
January 2, 2016
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 field of practice is computer science.
I would have liked it if solutions where available for a larger amount of the exercises. Since there are a lot of them it would have been helpful if the author had marked a selected subset as being the most important ones.
I would have liked it if solutions where available for a larger amount of the exercises. Since there are a lot of them it would have been helpful if the author had marked a selected subset as being the most important ones.
December 14, 2013
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 it will make the following math books much more enjoyable to read!
* 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 it will make the following math books much more enjoyable to read!
October 13, 2022
This was pretty insightful as far as I'm concerned 🥀.
My main concern was chapter 3 on Proof writing but I went through the first two chapters and had a go at the exercises too which weren't too difficult for me (surprising because I'm typically unable to solve them in other math books I pick up). I skimmed through Mathematical Induction, which I will likely be returning to sometime in the future out of regret.
The book is thankfully pretty accessible and there are answers for starred exercises (but only for them. Kind of wonder why they're never there for them all).
My main concern was chapter 3 on Proof writing but I went through the first two chapters and had a go at the exercises too which weren't too difficult for me (surprising because I'm typically unable to solve them in other math books I pick up). I skimmed through Mathematical Induction, which I will likely be returning to sometime in the future out of regret.
The book is thankfully pretty accessible and there are answers for starred exercises (but only for them. Kind of wonder why they're never there for them all).
January 12, 2008
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.
June 28, 2016
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 see the difference in his understanding for reading and writing proofs.
All the techniques of proofs(except induction) are covered in chapter-3. Post that, book introduced other topics like relations and functions and employs proof techniques for proving theorem in these topics. It was a great way to demonstrate that techniques learned for writing proofs are independent of any area and can be applied anywhere in mathematics.
I loved the treatment of proof by contradiction and mathematical induction. Cracking the corresponding exercises was a very rewarding experience. In many proofs when no approach seems to be working, proof by contradiction comes to the rescue. Similarly power of proof by induction was on display in solving many humongous problems.
All exercises were ordered from easy to moderate preparing the reader along the way to learn writing proofs for easier to challenging ones. Many exercises are built on top of the theorems from earlier exercises. This is a good thing as it helped me in two ways: revising the older chapters and discovering errors in my proofs.
There were many exercises asking the reader if the given proof is correct. Many times proof looked correct but turned out wrong because of a conceptual mistake. This helped tremendously in clearing many misconceptions.
In most of the sections, author also explains about how he arrived at a solution which helped in understanding how to approach a problem.
Finally in the last chapter author picked up a relatively advanced topic and employs all the proof techniques learned. In this chapter author does not go into explaining the proof structure but writes in a mathematical rigour so that reader should be able to read those proofs and gets an overall idea about reading and writing proofs by giving more focus to the topic than the proof technique.
One small thing that could have been better is the treatment of empty sets. I got confused while solving many exercises and felt like missing on some concepts regarding empty sets specially while dealing with family of sets.
To summarise,
- Quantifiers are everywhere.
- Reading and writing proofs.
- Set theory
- Mathematical Induction
- Developed some understanding for how to approach a problem.
- Felt great in solving many problems.
Overall it was a great endeavour and an enriching experience.
Originally written on my blog
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 see the difference in his understanding for reading and writing proofs.
All the techniques of proofs(except induction) are covered in chapter-3. Post that, book introduced other topics like relations and functions and employs proof techniques for proving theorem in these topics. It was a great way to demonstrate that techniques learned for writing proofs are independent of any area and can be applied anywhere in mathematics.
I loved the treatment of proof by contradiction and mathematical induction. Cracking the corresponding exercises was a very rewarding experience. In many proofs when no approach seems to be working, proof by contradiction comes to the rescue. Similarly power of proof by induction was on display in solving many humongous problems.
All exercises were ordered from easy to moderate preparing the reader along the way to learn writing proofs for easier to challenging ones. Many exercises are built on top of the theorems from earlier exercises. This is a good thing as it helped me in two ways: revising the older chapters and discovering errors in my proofs.
There were many exercises asking the reader if the given proof is correct. Many times proof looked correct but turned out wrong because of a conceptual mistake. This helped tremendously in clearing many misconceptions.
In most of the sections, author also explains about how he arrived at a solution which helped in understanding how to approach a problem.
Finally in the last chapter author picked up a relatively advanced topic and employs all the proof techniques learned. In this chapter author does not go into explaining the proof structure but writes in a mathematical rigour so that reader should be able to read those proofs and gets an overall idea about reading and writing proofs by giving more focus to the topic than the proof technique.
One small thing that could have been better is the treatment of empty sets. I got confused while solving many exercises and felt like missing on some concepts regarding empty sets specially while dealing with family of sets.
To summarise,
- Quantifiers are everywhere.
- Reading and writing proofs.
- Set theory
- Mathematical Induction
- Developed some understanding for how to approach a problem.
- Felt great in solving many problems.
Overall it was a great endeavour and an enriching experience.
Originally written on my blog
June 8, 2020
Terrific book, #1 in my reread-if-you-die-and-find-yourself-born-again list. But as someone who's been using it for self studying (if that's relevant) I'd change the structure of exercises a bit. Sometimes, there are too many of them so I got bored at the second dozen. At the same time details of covered topics often got faded away multiple paragraphs later. So, I'd split them in such a way that the most challenging ones (numbers 20-26s) are included not at the end of the list right after the relevant paragraph but a few paragraphs later under "repetition" section. I found this strategy of studying the most effective: I paid all the necessary attention to every problem and refreshed the knowledge of things I started to forget.
January 6, 2021
I have the first edition which doesn't have solutions, but there are several internet strangers that have solved all the problem and showcase them freely online. It's somewhat repetitive but very useful for practicing various proof techniques. I recommend the latest edition (3rd at the time of writing this review) of the book because it has additional exercises and plenty of solutions at the back, it makes it easier to check if you doubt yourself. This is a great introduction to thinking in proofs and showcasing your mental process neatly and correctly.
October 10, 2019
A book that teaches you how to construct well-typed formulas. Still, in my opinion, very far away from being a textbook that can teaches you the basics of set theory, proving and/or infinite sets. Use some other book instead, and read this book as a complementary source.
November 3, 2010
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 http://www.tricki.org/ %)
И к сожалению, этой книги почему-то нет на русском.
P. S.: еще жаль, что в универе я не знала про сайт http://www.tricki.org/ %)
January 15, 2017
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.
August 26, 2022
Good overview of logic, discrete math, and proof-writing concepts. Easy to follow.
December 21, 2020
This book is tedious, but ultimately worthwhile for the studious beginner that either wants or can stomach a lot of practice.
I was never that good at proofs in university. I bought this book to help. It didn't then, but has become more valuable to me as I gear up for round two of my struggle with mathematics.
There are three insights which are valuable in this book:
(1) All mathematics is built upon a hierarchy of logic and definitions, and so bigger terms can be broken down into smaller ones (functions are defined as predicates are defined as sets are defined as propositions are defined as terms with connectives, which together are true or false).
(2) Constructing proofs, rather than being mysterious, is instead a systematic process that can be done by applying the right rule to your statements over and over.
(3) Because this hierarchy is completely explicit and the rules are abundantly clear, if you break the statement down into its simplest parts and incrementally apply the right rules to them, you will be able to prove it.
What you should expect from this book is a path to inculcating these lessons. But if you have time, there are books I would recommend before this, and books I would recommend afterwards.
An alternative book is "Introduction to Mathematical Thinking" by Keith Devlin. It is more pedagogical than How to Prove It, as it takes time to explain the role precise language plays in mathematics, thus explaining why certain rules are the way they are (including why you negate the second part of the sentence rather than the first part, or why something false can imply that something else is true but not the other way around). As a bonus over Velleman, there is a Coursera course to accompany the book.
The book that made the mathematics of proof actually click for me was "A Logical Approach to Discrete Math" by Gries and Schneider. They teach a different but related method of proof called equational logic. It's worth bringing up because, they found that some students learn better with this style, as it's almost the same as simplifying arithmetic functions like we did in grade school, and it's even more mechanical than Velleman's givens-and-goals approach. Velleman also implicitly uses this approach when he simplifies his expressions before evaluating them. However, ALAtDM is in some ways the harder book, as it doesn't come with solutions and is very concise in its exposition. You can still find solutions online if you need them, and if you need more feedback to get started I would recommend this as a second book.
Honorable mention goes to "How to Read and Do Proofs" by Daniel Solow which can be seen as a summary of gloss of Velleman's book, even if written by a different author. Solow organizes his book around the so-called "Forward-Backward Method", which is equal to Velleman's approach to using givens and goals to find ways of finishing the proof in the middle. This makes "How to Prove It" a superset of Solow, but if you have less time on your hands then Solow might be good enough. Note: unlike Velleman, "How to Read and Do Proofs" does not cover set theory and predicate logic, so if that is important to you then pick Velleman.
I was never that good at proofs in university. I bought this book to help. It didn't then, but has become more valuable to me as I gear up for round two of my struggle with mathematics.
There are three insights which are valuable in this book:
(1) All mathematics is built upon a hierarchy of logic and definitions, and so bigger terms can be broken down into smaller ones (functions are defined as predicates are defined as sets are defined as propositions are defined as terms with connectives, which together are true or false).
(2) Constructing proofs, rather than being mysterious, is instead a systematic process that can be done by applying the right rule to your statements over and over.
(3) Because this hierarchy is completely explicit and the rules are abundantly clear, if you break the statement down into its simplest parts and incrementally apply the right rules to them, you will be able to prove it.
What you should expect from this book is a path to inculcating these lessons. But if you have time, there are books I would recommend before this, and books I would recommend afterwards.
An alternative book is "Introduction to Mathematical Thinking" by Keith Devlin. It is more pedagogical than How to Prove It, as it takes time to explain the role precise language plays in mathematics, thus explaining why certain rules are the way they are (including why you negate the second part of the sentence rather than the first part, or why something false can imply that something else is true but not the other way around). As a bonus over Velleman, there is a Coursera course to accompany the book.
The book that made the mathematics of proof actually click for me was "A Logical Approach to Discrete Math" by Gries and Schneider. They teach a different but related method of proof called equational logic. It's worth bringing up because, they found that some students learn better with this style, as it's almost the same as simplifying arithmetic functions like we did in grade school, and it's even more mechanical than Velleman's givens-and-goals approach. Velleman also implicitly uses this approach when he simplifies his expressions before evaluating them. However, ALAtDM is in some ways the harder book, as it doesn't come with solutions and is very concise in its exposition. You can still find solutions online if you need them, and if you need more feedback to get started I would recommend this as a second book.
Honorable mention goes to "How to Read and Do Proofs" by Daniel Solow which can be seen as a summary of gloss of Velleman's book, even if written by a different author. Solow organizes his book around the so-called "Forward-Backward Method", which is equal to Velleman's approach to using givens and goals to find ways of finishing the proof in the middle. This makes "How to Prove It" a superset of Solow, but if you have less time on your hands then Solow might be good enough. Note: unlike Velleman, "How to Read and Do Proofs" does not cover set theory and predicate logic, so if that is important to you then pick Velleman.
December 12, 2024
How to Prove It by Daniel Velleman is an outstanding introduction to mathematical proofs, ideal for anyone looking to deepen their understanding of logic and proof techniques. Even a middle schooler with a solid grasp of algebra could dive into this book and walk away with a wealth of knowledge. Velleman does an excellent job of breaking down complex ideas into manageable, digestible pieces, making it accessible to readers who are new to formal mathematical reasoning.
The book introduces fundamental topics like logic, set theory, and relations before moving into more advanced techniques such as direct proofs, proof by contradiction, and induction. Velleman’s approach is methodical and clear, guiding readers through each concept with well-organized explanations and plenty of examples. This thoughtful progression ensures that students not only learn the mechanics of proofs but also how to think logically about them, which is an invaluable skill in mathematics and beyond.
What makes this book stand out is its ability to make abstract topics feel approachable and engaging. The exercises are designed to build confidence gradually, making it an ideal starting point for anyone wanting to develop their proof-writing skills. It’s rare to find a textbook that manages to be both comprehensive and beginner-friendly, but How to Prove It strikes that balance perfectly.
For anyone serious about learning the art of proof, this book should easily earn a five-star rating. It’s a clear, concise, and incredibly valuable resource that not only teaches the foundational techniques of proofs but also helps readers build a deeper, more logical understanding of mathematics. Whether you're a high school student or just someone looking to sharpen your logical thinking, Velleman’s book is an excellent place to start.
The book introduces fundamental topics like logic, set theory, and relations before moving into more advanced techniques such as direct proofs, proof by contradiction, and induction. Velleman’s approach is methodical and clear, guiding readers through each concept with well-organized explanations and plenty of examples. This thoughtful progression ensures that students not only learn the mechanics of proofs but also how to think logically about them, which is an invaluable skill in mathematics and beyond.
What makes this book stand out is its ability to make abstract topics feel approachable and engaging. The exercises are designed to build confidence gradually, making it an ideal starting point for anyone wanting to develop their proof-writing skills. It’s rare to find a textbook that manages to be both comprehensive and beginner-friendly, but How to Prove It strikes that balance perfectly.
For anyone serious about learning the art of proof, this book should easily earn a five-star rating. It’s a clear, concise, and incredibly valuable resource that not only teaches the foundational techniques of proofs but also helps readers build a deeper, more logical understanding of mathematics. Whether you're a high school student or just someone looking to sharpen your logical thinking, Velleman’s book is an excellent place to start.
September 24, 2017
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.
This book is very accessible and demands from the student little in the way of prerequisite math knowledge.
August 31, 2024
Love this book. Superb intro to proofs.
December 24, 2024
Really good book to learn about fundamental math for higher courses in pure math.
January 31, 2019
La exposición de Velleman es sublime. Cada página de este libro es reveladora y es el comienzo para quien quiera profundizar en las matematicas. Creo, sin duda, que este libro debe ser utilizado por cada docente de matemáticas, pues en realidad ofrece una estructura sólida sobre como resolver un problema. El capítulo 4 en especial es una joya, sobre todo porque el autor te lleva de la mano para que descubras la lógica que hay detras de demostraciones matemáticas que parecerian complicadas y, sin embaro, te muestra paso por paso como se fue dando el resultado.
Hasta el capitulo 6, se tiene buen material para tomar cualquier libro de analisis matematico, teoria de numeros, topologia o bien algebra abstracta y entenderlo sin muchos problemas. El capitulo 7, sin embargo, va mas alla y contiene un tratamiento adecuado sobre conjuntos infinitos.
Sin duda recomiendo este libro para quien se inicie en las matematicas o, como un servidor, sea un aficionado sumamente interesado en profundizar en las raices de las matematicas.
Hasta el capitulo 6, se tiene buen material para tomar cualquier libro de analisis matematico, teoria de numeros, topologia o bien algebra abstracta y entenderlo sin muchos problemas. El capitulo 7, sin embargo, va mas alla y contiene un tratamiento adecuado sobre conjuntos infinitos.
Sin duda recomiendo este libro para quien se inicie en las matematicas o, como un servidor, sea un aficionado sumamente interesado en profundizar en las raices de las matematicas.
November 16, 2022
I had seen this book being recommended on math forums for people interested in honing their proof-writing skills. I certainly fitted the category, so I decided to give this a try.
Mr. Velleman does not disappoint with this beautiful exposition of the art of writing proofs. It's clear, didactic and very entertaining.
Maybe it has something to do with how the concept of a mathematical proof strikes me as beautiful, but throughout the entirebook I couldn't help but smile and be in awe at the sublime art which is mathematics.
I can't, however, rate this 5 stars due to what I considered a somewhat lacking explanation of set-related concepts. Once you get to the chapter on relations or infinite sets, the shaky foundations laid haphazardly on the early chapters clearly shows. It's a shame because I was excited to dig into the Cantor-Schröder-Bernstein theorem. Alas, one can't have everything in life.
Solid book, worth a careful read.
Mr. Velleman does not disappoint with this beautiful exposition of the art of writing proofs. It's clear, didactic and very entertaining.
Maybe it has something to do with how the concept of a mathematical proof strikes me as beautiful, but throughout the entirebook I couldn't help but smile and be in awe at the sublime art which is mathematics.
I can't, however, rate this 5 stars due to what I considered a somewhat lacking explanation of set-related concepts. Once you get to the chapter on relations or infinite sets, the shaky foundations laid haphazardly on the early chapters clearly shows. It's a shame because I was excited to dig into the Cantor-Schröder-Bernstein theorem. Alas, one can't have everything in life.
Solid book, worth a careful read.
October 5, 2019
My decision to work through this book was primarily so I could review proofwriting, and to this end Velleman's /How to Prove It/ was ideal for me for pretty much one reason - there are an impressive number of exercises.
The writing is clear, and I would recommend it to someone who had never encountered proofs before. The use I got out of it was more about practicing proofs rather than learning how to do them, but whenever I forgot something basic I could just go back and look and the explanations were not terribly bogged down by jargon.
The writing is clear, and I would recommend it to someone who had never encountered proofs before. The use I got out of it was more about practicing proofs rather than learning how to do them, but whenever I forgot something basic I could just go back and look and the explanations were not terribly bogged down by jargon.
January 18, 2021
How to Prove It is another "textbook" in a list of books that I wish I had read as a college student. Now, nearly a decade after graduation, I'm going back and getting the education I wish I would have given myself when I was younger, by studying books like this and Gujarati's Basic Econometrics.
The title of the book describes its purpose entirely: How to Prove It provides an in-depth course on how to write proofs. Anyone who plans to take college-level math courses (beyond calculus), and wants to actually succeed in them, should read this book.
The title of the book describes its purpose entirely: How to Prove It provides an in-depth course on how to write proofs. Anyone who plans to take college-level math courses (beyond calculus), and wants to actually succeed in them, should read this book.
September 17, 2023
I strongly recommend that high school students consider delving into this book to gain exposure to this unique style of thinking. While some reviewers who appreciated this book have raised a caveat regarding the perceived shortcomings of the chapter on set theory, I hold a nuanced perspective on this matter. In contrast to "Book of Proof," it is indeed true that less emphasis is placed on set theory within this work. Nevertheless, I personally found Velleman's treatment of the topic to be highly valuable in its application to the realm of probability theory.
November 30, 2019
The author is very patient in explaining the details to readers, but sometimes it gets too lengthy and confusing. The content is correct and rigorous, but there are some small inaccuracies (notation typos or nuances, in second edition). Overall, this is a good book to start getting familiar with mathematical proofs without too much intimidation of reading a full proof by oneself. It would be better if some parts get more concise.
July 24, 2020
Solid introduction
A solid introduction to - and survey of - logical proofs in mathematics. Covers propositional and first order predicate logic, mathematical induction and basic concepts of set theory. Lots of detailed examples. A few too many really. Time is limited!
The ebook formatting needs to be improved. It makes it just that bit harder to read. Excellent for beginners. A useful review/reminder for others.
A solid introduction to - and survey of - logical proofs in mathematics. Covers propositional and first order predicate logic, mathematical induction and basic concepts of set theory. Lots of detailed examples. A few too many really. Time is limited!
The ebook formatting needs to be improved. It makes it just that bit harder to read. Excellent for beginners. A useful review/reminder for others.
October 25, 2020
This book has been a tremendous help (and still is!) in preparing for studying math at university. Many lecturers basically skip over the proof techniques the author introduces in detail in this book. Thanks to the great exercises that are at the end of each chapter, I have so much less trouble with really hard and abstract exercises in our regular textbook because I have learned to look for the specific structures that the author explains.
July 13, 2020
It was my textbook on Discrete Math course.
It's a valuable book mostly for the fact it's actually teaches you how to write a well constructed proofs.
Each chapter reveals its genius proofs with behind a scene strategies applied on it and lots of examples demonstrating the process pf proof writing.
Also the book includes a challenging exercises. Highly recommended!
It's a valuable book mostly for the fact it's actually teaches you how to write a well constructed proofs.
Each chapter reveals its genius proofs with behind a scene strategies applied on it and lots of examples demonstrating the process pf proof writing.
Also the book includes a challenging exercises. Highly recommended!
May 15, 2022
Eu li isso aí em 2012, por recomendação do /sci/. Bom livro, foi a minha primeira introdução a demonstrações matemáticas. Saudades de usar o proof designer para provar trocentas propriedades de conjuntos pela definição sem poder reutilizar resultados já obtidos...
September 4, 2019
A wonderful book that introduces practical strategies for proving different things in mathematics, and makes all kinds of proofs seem manageable to a beginning student, no small achievement!
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January 6, 2021
best books for foundations
December 13, 2021
Good reading material, but can seem very common-sense. Maybe it will help with your intuition, but a discrete maths textbook with lots of practice problems will probably help more.
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