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 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

# 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 brea 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

Compare

5out of 5Simon Vindum–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 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.

4out of 5Jessica Austin–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 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!

5out of 5Honk Honkerson–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 proo 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.

5out of 5Achmed–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. 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.

4out of 50vai5–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 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

4out of 5Benjamin Schneider–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.

4out of 5Anthony James–I should have read something like this years ago, at the end of secondary school or start of university. I intend on returning to it as I skipped more and more of the exercises as I moved through the text. Highly recommend to people who don't like proofs! I should have read something like this years ago, at the end of secondary school or start of university. I intend on returning to it as I skipped more and more of the exercises as I moved through the text. Highly recommend to people who don't like proofs!

4out of 5Ege Onur–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.

5out of 5John Doe–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 re 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.

5out of 5Eryk Banatt–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 explanation 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.

5out of 5Nick Crowley–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 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.

4out of 5THN–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.

5out of 5Anthony O'Connor–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.

4out of 5Esther Theresia–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.

4out of 5Libby–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 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!

5out of 5Mahdi Dibaiee–A great book on mathematical proofs for someone with very limited prior knowledge. The exercises are great, though they are plenty and can take a considerable amount of time to work through.

5out of 5Henry Cooksley–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!

5out of 5Lukasz–recommendation: [Ask HN: How can I learn to read mathematical notation? | Hacker News](https://news.ycombinator.com/item?id=...) recommendation: [Ask HN: How can I learn to read mathematical notation? | Hacker News](https://news.ycombinator.com/item?id=...)

4out of 5Liam Tarr–Equally appropriate for the advanced high school student or the student in the first two years of their undergraduate studies. A great antidote to proof anxiety.

4out of 5Shubham Kumar–best books for foundations

5out of 5Kenneth B–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 (fun 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.

4out of 5Daeus–I love how this book was structured, especially the first half (which is where I had most of my 'ah-ha!' moments). Each section built off the previous one masterfully. It filled in a lot of gaps for me in my mathematics and understanding of logic, even after majoring in math for undergrad. It took me well over a year to get through and though definitely brutal at times I am so grateful for these learnings. I gave up trying to do the exercises a few chapters after the proof section since it becam I love how this book was structured, especially the first half (which is where I had most of my 'ah-ha!' moments). Each section built off the previous one masterfully. It filled in a lot of gaps for me in my mathematics and understanding of logic, even after majoring in math for undergrad. It took me well over a year to get through and though definitely brutal at times I am so grateful for these learnings. I gave up trying to do the exercises a few chapters after the proof section since it became more about just reading the proofs and less about mastering the techniques. I think working in programming/analytics after college really helped my understand of logical syntax and proof concepts that I struggled so much with in college. My main gaps were really sentential logic, reading comprehension/patience, and a lack of understanding of some basic definitions (ie existence, uniqueness, onto, contrapositive, etc). Quotes: - "To make sure your assertions are adequately justified, you must be skeptical about every inference in your proof. If there is any doubt in your mind about whether the justification you have given for an assertion is adequate, the it isn't. After all, if your own reasoning doesn't even convince you, how can you expect it to convince anybody else?" - "When mathematicians write proofs, they usually just write the steps needed to justify their conclusions with no explanation of how they thought of them.... Although this lack of explanation sometimes makes proofs hard to read, it serves the purpose of keeping two distinct objectives separate: explaining your thought process and justifying your conclusions. The first is psychology, the second is mathematics....Occasionally, in a very complicated proof, a mathematician may include some discussion of the strategy behind the proof to make the proof easier to read. Usually, however, it is up to readers to figure this out themselves." This explains so much.... - "it's important to stick to the rules of proof-writing we've studied rather than allow yourself to be convinced by any reasoning that looks plausible....It is only by turning your idea into a formal proof that you can be sure your answer is right. Often in the course of trying to construct a formal proof you will discover a flaw in your reasoning...and you may have to overcome the flaw. The final theorem and proof are often the result of repeated mistakes and corrections....just because mathematicians don't explain their mistakes in their proofs, you shouldn't be fooled into thinking they don't make any!"

4out of 5Manuel–"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 "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 the book cover-to-cover without pausing; but I intend on going back to the beginning and really work through the many (many!) exercises in the text. The book is destined to become a classic. I highly recommend it!

5out of 5Andi–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 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 many, or as little as you need. This is where I think the real strength of the book lies. The back contains several solutions, but you can find the rest of it online through some googling.

5out of 5Andre Harmse–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 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 exercises and many suggestions and full solutions in the back of the book.

4out of 5William Schram–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.

4out of 5Chris Ereth–"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. "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.

4out of 5Reinier Tromp–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.

4out of 5MLO–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. 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.

5out of 5Kevin Montes–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