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How to Prove It: A Structured Approach

Description:

Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of 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 an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians.
Read more

Editorial Reviews

Review

'Not only does this book help students learn how to prove results, it highlights why we care so much. It starts in the introduction with some simple conjectures and gathering data, quickly disproving the first but amassing support for the second. Will that pattern persist? How can these observations lead us to a proof? The book is engagingly written, and covers - in clear and great detail - many proof techniques. There is a wealth of good exercises at various levels. I've taught problem solving before (at The Ohio State University and Williams College), and this book has been a great addition to the resources I recommend to my students.' Steven J. Miller, Williams College, Massachusetts

'This book is my go-to resource for students struggling with how to write mathematical proofs. Beyond its plentiful examples, Velleman clearly lays out the techniques and principles so often glossed over in other texts.' Rafael Frongillo, University of Colorado, Boulder

'I've been using this book religiously for the last eight years. It builds a strong foundation in proof writing and creates the axiomatic framework for future higher-level mathematics courses. Even when teaching more advanced courses, I recommend students to read chapter 3 (Proofs) since it is, in my opinion, the best written exposition of proof writing techniques and strategies. This third edition brings a new chapter (Number Theory), which gives the instructor a few more topics to choose from when teaching a fundamental course in mathematics. I will keep using it and recommending it to everyone, professors and students alike.' Mihai Bailesteanu, Central Connecticut State University

'Professor Velleman sets himself the difficult task of bridging the gap between algorithmic and proof-based mathematics. By focusing on the basic ideas, he succeeded admirably. Many similar books are available, but none are more treasured by beginning students. In the Third Edition, the constant pursuit of excellence is further reinforced.' Taje Ramsamujh, Florida International University

'Proofs are central to mathematical development. They are the tools used by mathematicians to establish and communicate their results. The developing mathematician often learns what constitutes a proof and how to present it by osmosis. How to Prove It aims at changing that. It offers a systematic introduction to the development, structuring, and presentation of logical mathematical arguments, i.e. proofs. The approach is based on the language of first-order logic and supported by proof techniques in the style of natural deduction. The art of proving is exercised with naive set theory and elementary number theory throughout the book. As such, it will prove invaluable to first-year undergraduate students in mathematics and computer science.' Marcelo Fiore, University of Cambridge

'Overall, this is an engagingly-written and effective book for illuminating thinking about and building a careful foundation in proof techniques. I could see it working in an introduction to proof course or a course introducing discrete mathematics topics alongside proof techniques. As a self-study guide, I could see it working as it so well engages the reader, depending on how able they are to navigate the cultural context in some examples.' Peter Rowlett, LMS Newsletter

‘Altogether this is an ambitious and largely very successful introduction to the writing of good proofs, laced with many good examples and exercises, and with a pleasantly informal style to make the material attractive and less daunting than the length of the book might suggest. I particularly liked the many discussions of fallacious or incomplete proofs, and the associated challenges to readers to untangle the errors in proofs and to decide for themselves whether a result is true.’ Peter Giblin, University of Liverpool, The Mathematical Gazette

Book Description

Helps students transition from problem solving to proving theorems, with a new chapter on number theory and over 150 new exercises.

Reviews:

5.0 out of 5 stars Math coach

Y. · September 26, 2025

A good brief guide to learning

5.0 out of 5 stars Nice

J.B. · January 23, 2025

Nice structure overall, good quality and great illustrations.

5.0 out of 5 stars Very good book

I.W. · July 30, 2024

This book is very well done and is quite engaging. It has good practice problems and is really quite thorough. Overall a very solid choice for learning proofs.

4.0 out of 5 stars Fantastique

K.P. · August 2, 2020

If you study this book well, you will become highly skilled in doing mathematical proofs, but not just. After having only gone thoroughly through the first chapters, you will be so skilled that you can skip the material at the beginning of many math textbooks that review set theory, etc. For instance, Munkres, Topology. The material at the beginning of the book becomes an utter triviality. Your understanding of proofs in Real Analysis textbooks will be ameliorated. I can not fully explain how this will help you.I personally read How to Read and Do Proofs, Solow, but after going through that textbook then picking up this one. I would advise you to not bother with Solow's text. He makes up his own terminology for things that already exist, and it's kind of handwavy. You will find that you have to relearn what you're doing if you read his, but you will see where he was coming from on his techniques. His techniques are wrong, but there is a better way. Use this book. I never give anything 5 stars. 4 is the max I rate out of 5; it would be 9 stars, if it were up to 10, etc.Buy this book, study it, go back to the beginning of the book, and review it, just like you would if you were taking a college exam on the material. Ensure your mastery. You will not regret it.

5.0 out of 5 stars Amazing introduction to mathematical reasoning

I.U. · February 10, 2020

Wow, wow, wow. This is definitely the best math book I have ever read. I am currently starting chapter 3, and have worked through all the examples and exercises so far. This book covers everything one should learn in a discrete math/introduction to proofs course and a little bit more.There are plenty of examples, and the exercises are very accessible while still being nontrivial. There are solutions to some problems in the back of the book, but many of the exercises are written in such a way that you can verify the answers yourself.Physically, the book is flawless. Extremely high quality pages, large font size, and a smaller frame closer to the size of a novel.

5.0 out of 5 stars One of my favorites books in mathematics

O.o.m.f. · August 24, 2023

It's a great introduction for math proofs, and not only that, it has a wide variety of exercises some of them pretty challenging. It's a great way to study and learn how to write mathematical proofs

5.0 out of 5 stars Awsome book

J. · December 28, 2023

Just as it stated, It allows a structured approach to problem solving.

5.0 out of 5 stars A wonderful introduction to "real" math

A.C. · November 2, 2019

This is a remarkable book! It focuses narrowly on mathematical logic, set theory, and the application of both to theorem-proving. In practice this works very well, provided the reader is willing to read the text and the examples carefully, and provided that he or she is willing to put the work into the provided exercises. This is the most useful math book I have ever opened, and I think it's rather accessible for what it is.

Advanced math

J. · September 7, 2024

Quite challenging!!

Do we need truth tables?

C.K. · July 13, 2025

Truth tables are a pain in the arch

Bom

E.H.N. · July 25, 2025

Bom

Great book for beginning math proof writing

R. · October 13, 2023

Used this book to supplement a college math course.Explanations are very clear and to the point with ample examples. Book is a perfect size for reading in transit. Only wished that more solutions are provided in the exercises.

Awesome book. Get it if you need help with proofs!

W.C. · December 30, 2020

I am almost done the first chapter and I am really happy with the quality of this book. I would buy this book a thousand times again if I had to. Totally worth every cent.

Visit the Cambridge University Press Store

How to Prove It: A Structured Approach

Product ID: U1108439535
Condition: New

4.7

AED29421

Price includes VAT & Import Duties
Type: Paperback
Availability: In Stock

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Visit the Cambridge University Press Store

How to Prove It: A Structured Approach

Product ID: U1108439535
Condition: New

4.7

How to Prove It: A Structured Approach-0
Type: Paperback

AED29421

Price includes VAT & Import Duties
Availability: In Stock

Quantity:

|

Order today to get by 7-14 business days

This item qualifies for free delivery

Returns & Warranty policies

Imported From: United States

At BOLO, we work hard to ensure the products you receive are new, genuine, and sourced from reputable suppliers.

BOLO is not an authorized or official retailer for most brands, nor are we affiliated with manufacturers unless specifically stated on a product page. Instead, we source verified sellers, authorized distributors or directly from the manufacturer.

Each product undergoes thorough inspection and verification at our consolidation and fulfilment centers to ensure it meets our strict authenticity and quality standards before being shipped and delivered to you.

If you ever have concerns regarding the authenticity of a product purchased from us, please contact Bolo Support. We will review your inquiry promptly and, if necessary, provide documentation verifying authenticity or offer a suitable resolution.

Your trust is our top priority, and we are committed to maintaining transparency and integrity in every transaction.

All product information, images, descriptions, and reviews originate from the manufacturer or from trusted sellers overseas. BOLO is not affiliated with, endorsed by, or an authorized retailer for most brands listed on our website unless stated otherwise.

While we strive to display accurate information, variations in packaging, labeling, instructions, or formulation may occasionally occur due to regional differences or supplier updates. For detailed or manufacturer-specific information, please contact the brand directly or reach out to BOLO Support for assistance.

Unless otherwise stated, all prices displayed on the product page include applicable taxes and import duties.

BOLO operates in accordance with the laws and regulations of United Arab Emirates. Any items found to be restricted or prohibited for sale within the UAE will be cancelled prior to shipment. We take proactive measures to ensure that only products permitted for sale in United Arab Emirates are listed on our website.

All items are shipped by air, and any products classified as “Dangerous Goods (DG)” under IATA regulations will be removed from the order and cancelled.

All orders are processed manually, and we make every effort to process them promptly once confirmed. Products cancelled due to the above reasons will be permanently removed from listings across the website.

Description:

Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of 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 an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians.
Read more

Editorial Reviews

Review

'Not only does this book help students learn how to prove results, it highlights why we care so much. It starts in the introduction with some simple conjectures and gathering data, quickly disproving the first but amassing support for the second. Will that pattern persist? How can these observations lead us to a proof? The book is engagingly written, and covers - in clear and great detail - many proof techniques. There is a wealth of good exercises at various levels. I've taught problem solving before (at The Ohio State University and Williams College), and this book has been a great addition to the resources I recommend to my students.' Steven J. Miller, Williams College, Massachusetts

'This book is my go-to resource for students struggling with how to write mathematical proofs. Beyond its plentiful examples, Velleman clearly lays out the techniques and principles so often glossed over in other texts.' Rafael Frongillo, University of Colorado, Boulder

'I've been using this book religiously for the last eight years. It builds a strong foundation in proof writing and creates the axiomatic framework for future higher-level mathematics courses. Even when teaching more advanced courses, I recommend students to read chapter 3 (Proofs) since it is, in my opinion, the best written exposition of proof writing techniques and strategies. This third edition brings a new chapter (Number Theory), which gives the instructor a few more topics to choose from when teaching a fundamental course in mathematics. I will keep using it and recommending it to everyone, professors and students alike.' Mihai Bailesteanu, Central Connecticut State University

'Professor Velleman sets himself the difficult task of bridging the gap between algorithmic and proof-based mathematics. By focusing on the basic ideas, he succeeded admirably. Many similar books are available, but none are more treasured by beginning students. In the Third Edition, the constant pursuit of excellence is further reinforced.' Taje Ramsamujh, Florida International University

'Proofs are central to mathematical development. They are the tools used by mathematicians to establish and communicate their results. The developing mathematician often learns what constitutes a proof and how to present it by osmosis. How to Prove It aims at changing that. It offers a systematic introduction to the development, structuring, and presentation of logical mathematical arguments, i.e. proofs. The approach is based on the language of first-order logic and supported by proof techniques in the style of natural deduction. The art of proving is exercised with naive set theory and elementary number theory throughout the book. As such, it will prove invaluable to first-year undergraduate students in mathematics and computer science.' Marcelo Fiore, University of Cambridge

'Overall, this is an engagingly-written and effective book for illuminating thinking about and building a careful foundation in proof techniques. I could see it working in an introduction to proof course or a course introducing discrete mathematics topics alongside proof techniques. As a self-study guide, I could see it working as it so well engages the reader, depending on how able they are to navigate the cultural context in some examples.' Peter Rowlett, LMS Newsletter

‘Altogether this is an ambitious and largely very successful introduction to the writing of good proofs, laced with many good examples and exercises, and with a pleasantly informal style to make the material attractive and less daunting than the length of the book might suggest. I particularly liked the many discussions of fallacious or incomplete proofs, and the associated challenges to readers to untangle the errors in proofs and to decide for themselves whether a result is true.’ Peter Giblin, University of Liverpool, The Mathematical Gazette

Book Description

Helps students transition from problem solving to proving theorems, with a new chapter on number theory and over 150 new exercises.

Reviews:

5.0 out of 5 stars Math coach

Y. · September 26, 2025

A good brief guide to learning

5.0 out of 5 stars Nice

J.B. · January 23, 2025

Nice structure overall, good quality and great illustrations.

5.0 out of 5 stars Very good book

I.W. · July 30, 2024

This book is very well done and is quite engaging. It has good practice problems and is really quite thorough. Overall a very solid choice for learning proofs.

4.0 out of 5 stars Fantastique

K.P. · August 2, 2020

If you study this book well, you will become highly skilled in doing mathematical proofs, but not just. After having only gone thoroughly through the first chapters, you will be so skilled that you can skip the material at the beginning of many math textbooks that review set theory, etc. For instance, Munkres, Topology. The material at the beginning of the book becomes an utter triviality. Your understanding of proofs in Real Analysis textbooks will be ameliorated. I can not fully explain how this will help you.I personally read How to Read and Do Proofs, Solow, but after going through that textbook then picking up this one. I would advise you to not bother with Solow's text. He makes up his own terminology for things that already exist, and it's kind of handwavy. You will find that you have to relearn what you're doing if you read his, but you will see where he was coming from on his techniques. His techniques are wrong, but there is a better way. Use this book. I never give anything 5 stars. 4 is the max I rate out of 5; it would be 9 stars, if it were up to 10, etc.Buy this book, study it, go back to the beginning of the book, and review it, just like you would if you were taking a college exam on the material. Ensure your mastery. You will not regret it.

5.0 out of 5 stars Amazing introduction to mathematical reasoning

I.U. · February 10, 2020

Wow, wow, wow. This is definitely the best math book I have ever read. I am currently starting chapter 3, and have worked through all the examples and exercises so far. This book covers everything one should learn in a discrete math/introduction to proofs course and a little bit more.There are plenty of examples, and the exercises are very accessible while still being nontrivial. There are solutions to some problems in the back of the book, but many of the exercises are written in such a way that you can verify the answers yourself.Physically, the book is flawless. Extremely high quality pages, large font size, and a smaller frame closer to the size of a novel.

5.0 out of 5 stars One of my favorites books in mathematics

O.o.m.f. · August 24, 2023

It's a great introduction for math proofs, and not only that, it has a wide variety of exercises some of them pretty challenging. It's a great way to study and learn how to write mathematical proofs

5.0 out of 5 stars Awsome book

J. · December 28, 2023

Just as it stated, It allows a structured approach to problem solving.

5.0 out of 5 stars A wonderful introduction to "real" math

A.C. · November 2, 2019

This is a remarkable book! It focuses narrowly on mathematical logic, set theory, and the application of both to theorem-proving. In practice this works very well, provided the reader is willing to read the text and the examples carefully, and provided that he or she is willing to put the work into the provided exercises. This is the most useful math book I have ever opened, and I think it's rather accessible for what it is.

Advanced math

J. · September 7, 2024

Quite challenging!!

Do we need truth tables?

C.K. · July 13, 2025

Truth tables are a pain in the arch

Bom

E.H.N. · July 25, 2025

Bom

Great book for beginning math proof writing

R. · October 13, 2023

Used this book to supplement a college math course.Explanations are very clear and to the point with ample examples. Book is a perfect size for reading in transit. Only wished that more solutions are provided in the exercises.

Awesome book. Get it if you need help with proofs!

W.C. · December 30, 2020

I am almost done the first chapter and I am really happy with the quality of this book. I would buy this book a thousand times again if I had to. Totally worth every cent.

Similar suggestions by Bolo

More from this brand

Similar items from “Logic”

Share with

Or share with link

https://www.bolo.ae/products/U1108439535