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Book of Proof
This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity. Topics include sets, logic, counting, methods of conditional and non-conditional proof, disproof, induction, relations, functions and infinite cardinality.
303 pages, Paperback
First published January 1, 2009
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Displaying 1 - 30 of 36 reviews
October 18, 2016
Don't try to use this book to win an argument against your wife.
February 7, 2016
Outstanding. I have used this as a supplemental text in a few courses, and would totally consider adopting it as the primary text for the right course. There is certainly enough here for a full semester's work. And, it's open source, which means it is not only free (the author's website is a google away) but also remixable - meaning, you are free to create and distribute your own version of the text under a creative commons license (http://creativecommons.org/licenses/b...) as long as you do so under the same license as the original.
Very little is assumed of the reader at the beginning, which surprisingly is not a given for such a text. There is a potentially useful chapter on combinatorics - basic stuff (multiplication principle, factorials, Pascal's triangle, a little inclusion-exclusion) that we can't necessarily assume our students have seen in high school. Standard techniques including proof by contradiction and mathematical induction are well-represented. The material on cardinality (at the end of the book) is nice.
Two things stand out about Book of Proof, compared not only to other free books, but compared to any other books for a "proofs" or "transitions" course: first, the proofs presented herein are not all presented in a finished form; Hammack develops them in a way that reflects the thought process. Second, there are tons of completely written-out proofs in the answers section at the back.
I contend that these are very significant things, and that they are sufficient for anyone teaching a course of this nature to consider adopting this text. I see no reason not to at least add it as a supplemental text - not just in the "transitions" course, but for other courses where students will be struggling with the same issues, eg. Discrete Math.
Very little is assumed of the reader at the beginning, which surprisingly is not a given for such a text. There is a potentially useful chapter on combinatorics - basic stuff (multiplication principle, factorials, Pascal's triangle, a little inclusion-exclusion) that we can't necessarily assume our students have seen in high school. Standard techniques including proof by contradiction and mathematical induction are well-represented. The material on cardinality (at the end of the book) is nice.
Two things stand out about Book of Proof, compared not only to other free books, but compared to any other books for a "proofs" or "transitions" course: first, the proofs presented herein are not all presented in a finished form; Hammack develops them in a way that reflects the thought process. Second, there are tons of completely written-out proofs in the answers section at the back.
I contend that these are very significant things, and that they are sufficient for anyone teaching a course of this nature to consider adopting this text. I see no reason not to at least add it as a supplemental text - not just in the "transitions" course, but for other courses where students will be struggling with the same issues, eg. Discrete Math.
June 23, 2018
Playing with Numbers
22 June 2018 - Sydney
Well, what do you know, a university textbook that doesn’t actually break the bank. In fact a University text book that actually costs you absolutely nothing. I’m definitely going to give this book some good marks for that. Yeah, I remember the last time I was at Uni and the most expensive aspect of it was the books (well, yes, the fees, but the government technically pays for that). In fact when I did Law you could be assured that the yearly book bill would run into the hundreds of dollars. Arts wasn’t as bad, since half the subjects I did involved reading novels (though they would always want you to read a specific edition of the novel, namely an edition that the university would get a kickback for if a copy was sold, and there were those text books that the lecturers would write, and then conveniently put on the reading list every semester). Well, not so with this book, or at least the digital download edition, so well played.
Now, as one reviewer pointed out, a book entitled ‘The Book of Proof’ isn’t the type of book that is going to assist you with winning an argument with your wife because, well, that’s never going to happen. In fact, this isn’t even going to help you win an argument with an idiot because, well, this book relies upon arguing logically, and since when did idiots follow logic. In fact since when did idiots actually follow any structural form in their arguments, except for attempting to shut you down through the use of some fallacy, and then proceed to employ that fallacy in their own arguments. No wonder I simply don’t have time for some people.
Anyway, this is actually a mathematic text book and is all about how to do mathematical proofs. Now, I actually don’t mind maths all that much, namely because once you understand how to do something the numbers literally work themselves out. Well, sort of, I’m probably more of a pure maths guy as opposed to an applied maths guy, namely because when it comes to applied maths you sort of have to not only find the numbers, but also work out where to plug them. Fortunately this isn’t strictly applied maths but rather it falls into a category of maths call discrete maths.
So, once you had thought you pretty much had a good grasp on anything and everything mathematical you suddenly land up at your first day at uni only to discover that the maths you thought you know has become incredibly insane. For instance, graphs aren’t what you thought they were in High School maths, now you encounter a completely different kind of graph, and also discover that mathematicians plant their trees upside down. Oh, and don’t get me started on Infinity. You know, I always thought that infinity meant ‘to go on forever’, or ‘to be uncountable’. Well, it turns out that there is more than one type of infinity, and you can have a ‘countable infinity’.
Like seriously, more than one! That’s insane. Well, that’s what all the mathematicians told Georg Cantor when he first proposed the idea, and he ended up dying in a mental asylum. Yeah, seriously, maths really does mess with your head. Then again, if Cantor had read his history, he would have known all about this because everything somebody attempting to break the mold and introduce something a little different, there would be a huge reaction. You know, like the number zero, which European mathematicians avoided because it was, well, an Arab concept. Yeah, they were like that, but then again this was why they called it the Dark Ages, particularly since arguments centered around the idea of how many angels could dance on the head of a pin (I’m going to stick with one).
So, I’m not entirely sure how this is going to help you win your next argument, unless of course that argument is of a mathematical nature, and you are arguing whether the square root of two is irrational or not, but then again that’s sort of an argument that you shouldn’t be having because if somebody tries to argue otherwise basically has absolutely no idea what they are talking about, and it doesn’t matter how much mathematical proof you throw at them, if they don’t realise that the square root of two is irrational, then it’s probably not one of those conversations that you really should be having in the first place.
22 June 2018 - Sydney
Well, what do you know, a university textbook that doesn’t actually break the bank. In fact a University text book that actually costs you absolutely nothing. I’m definitely going to give this book some good marks for that. Yeah, I remember the last time I was at Uni and the most expensive aspect of it was the books (well, yes, the fees, but the government technically pays for that). In fact when I did Law you could be assured that the yearly book bill would run into the hundreds of dollars. Arts wasn’t as bad, since half the subjects I did involved reading novels (though they would always want you to read a specific edition of the novel, namely an edition that the university would get a kickback for if a copy was sold, and there were those text books that the lecturers would write, and then conveniently put on the reading list every semester). Well, not so with this book, or at least the digital download edition, so well played.
Now, as one reviewer pointed out, a book entitled ‘The Book of Proof’ isn’t the type of book that is going to assist you with winning an argument with your wife because, well, that’s never going to happen. In fact, this isn’t even going to help you win an argument with an idiot because, well, this book relies upon arguing logically, and since when did idiots follow logic. In fact since when did idiots actually follow any structural form in their arguments, except for attempting to shut you down through the use of some fallacy, and then proceed to employ that fallacy in their own arguments. No wonder I simply don’t have time for some people.
Anyway, this is actually a mathematic text book and is all about how to do mathematical proofs. Now, I actually don’t mind maths all that much, namely because once you understand how to do something the numbers literally work themselves out. Well, sort of, I’m probably more of a pure maths guy as opposed to an applied maths guy, namely because when it comes to applied maths you sort of have to not only find the numbers, but also work out where to plug them. Fortunately this isn’t strictly applied maths but rather it falls into a category of maths call discrete maths.
So, once you had thought you pretty much had a good grasp on anything and everything mathematical you suddenly land up at your first day at uni only to discover that the maths you thought you know has become incredibly insane. For instance, graphs aren’t what you thought they were in High School maths, now you encounter a completely different kind of graph, and also discover that mathematicians plant their trees upside down. Oh, and don’t get me started on Infinity. You know, I always thought that infinity meant ‘to go on forever’, or ‘to be uncountable’. Well, it turns out that there is more than one type of infinity, and you can have a ‘countable infinity’.
Like seriously, more than one! That’s insane. Well, that’s what all the mathematicians told Georg Cantor when he first proposed the idea, and he ended up dying in a mental asylum. Yeah, seriously, maths really does mess with your head. Then again, if Cantor had read his history, he would have known all about this because everything somebody attempting to break the mold and introduce something a little different, there would be a huge reaction. You know, like the number zero, which European mathematicians avoided because it was, well, an Arab concept. Yeah, they were like that, but then again this was why they called it the Dark Ages, particularly since arguments centered around the idea of how many angels could dance on the head of a pin (I’m going to stick with one).
So, I’m not entirely sure how this is going to help you win your next argument, unless of course that argument is of a mathematical nature, and you are arguing whether the square root of two is irrational or not, but then again that’s sort of an argument that you shouldn’t be having because if somebody tries to argue otherwise basically has absolutely no idea what they are talking about, and it doesn’t matter how much mathematical proof you throw at them, if they don’t realise that the square root of two is irrational, then it’s probably not one of those conversations that you really should be having in the first place.
November 1, 2021
Clear as crystal
July 31, 2022
A gentle introduction to sets, logic and methods of mathematical proof that's good for self study. It has lots of exercises and solutions to half of them at the back. The paperback is also ~$20 on Amazon (and the quality of the printing is not bad) and the PDF is free on the author's website. The combinatorics chapter is particularly good. My only complaint is that the exercises get rather rote (which is why I dropped the book halfway through so I can't speak for parts III and IV); repetition is important for getting students used to the formalisms, but I think including some more challenging exercises for students to tackle once they've gotten comfortable with the proof style would make the book more useful for a wider range of experience levels and prepare students better for further math courses.
March 26, 2015
I feel weird giving 5 stars to a book about proofs. It's not a great book in the sense that The Great Gatsby is. Getting over that, I'm assuming anyone who would look at this book would know what to expect from it.
It has a specific job, to teach basic proofs. At that job, it is the best book available. You can get the PDF for free on Prof. Hammack's website, but I bought the physical copy. So before you start your Real Analysis course with Baby Rudin or your graduate economics courses with proofs, work through this book. It will be well worth the effort.
If you have done many proofs, this book isn't for you. That's not the book's fault. So 5/5 stars it is.
It has a specific job, to teach basic proofs. At that job, it is the best book available. You can get the PDF for free on Prof. Hammack's website, but I bought the physical copy. So before you start your Real Analysis course with Baby Rudin or your graduate economics courses with proofs, work through this book. It will be well worth the effort.
If you have done many proofs, this book isn't for you. That's not the book's fault. So 5/5 stars it is.
March 18, 2021
Good for self studying.
August 29, 2017
Hammacks's Book of Proof is a wonderful introduction to the world of upper division mathematics. I think that this book would be perfect to read before starting with LA/DEs and I wish I had read it before taking those classes. Hammack gently leads the reader through many important mathematic concepts that are taken as understood by the reader in many upper division classes. This books is a must read for anyone going into upper division math, regardless of major!
July 29, 2018
It introduces the fundamental concepts of Discrete Mathematics and Proofs. You will learn the basic syntax languages used by Mathematicians.
The theories are explained with Proofs, described in a clear and concise way, with sufficient exercises at the end. I went thru the exercises, and took me awhile to finish, but it was enjoyable.
Topics include: Sets, Logic, Counting, Relations, Functions, Set Cardinality
Proof methods include: Direct, Contrapositive, Contradiction, Induction, etc.
The theories are explained with Proofs, described in a clear and concise way, with sufficient exercises at the end. I went thru the exercises, and took me awhile to finish, but it was enjoyable.
Topics include: Sets, Logic, Counting, Relations, Functions, Set Cardinality
Proof methods include: Direct, Contrapositive, Contradiction, Induction, etc.
July 7, 2025
I love this book for being really student-friendly. It was a good experience reading the book in my first semester. However i found some parts like some topics from 'relations' to be hard. I expect myself to improve my understanding as well as the writer to provide more explanation with easier examples. The overall experience with this book is simply amazing!
October 21, 2019
There are certain ways to write a mathematical proof. We all know that but the unfortunate thing is we are rarely taught that. This book provides the foundation which enables one to be confident about their proofs and write more rigorous proofs.
I will come back to this book again and again.
I will come back to this book again and again.
November 17, 2020
The perfect introduction to "real" math, not much to said here that hasn't already been said. Also you'll probably be forced to read this anyways if you're here so it's not like this review even matters to you haha
January 8, 2024
If there is a rustic landscape of mathematics, exuding calmness, this book is that one. Currently reading, yet already rated five. I wish I discovered this gem, open access, early in my sixth grade in stead of those self-regarding Bangladeshi Math Olympiad books.
March 25, 2020
Great book on the processes and methodologies of formal rigorous mathematical proofs. A must for a mathematics student(s).
April 17, 2020
A masterpiece on proofs. Focus on a set theory and logic. Amazing selection of exercises. Doing all exercises is a must
Read
October 20, 2020Really great.
I was looking for an introductory (really introductory) book on proof and I found it.
I was looking for an introductory (really introductory) book on proof and I found it.
December 4, 2020
Very interesting textbook. Easy to follow, I would definitely recommend. One of the best textbooks I’ve read in my academic career.
September 1, 2021
Beautifully written and structured. A great book.
March 15, 2023
This book changed my life by introducing me to mathematics.
it's an essential read for any programmer who isnt familiar with any of the theoretical foundations
it's an essential read for any programmer who isnt familiar with any of the theoretical foundations
August 4, 2023
I'll let this brief video be my review of this book and Discrete Math:
https://photos.app.goo.gl/S4gW3V5qmPR...
Copy and paste the link into your web browser and you can see my video review.
https://photos.app.goo.gl/S4gW3V5qmPR...
Copy and paste the link into your web browser and you can see my video review.
May 15, 2024
Good overview on proofs.
October 13, 2024
The easiest to understand of the proof books I tried. I tried long form, how to solve it, 100 % mathematical proof. But it is a little bit boring
January 5, 2025
the exercises are great.
January 9, 2025
Great for self study. Has a lot of examples with solutions in the back. Also available for free online.
January 28, 2025
This book got me into proofs and helped me a great deal in reading future math-based books.
February 7, 2025
soyut matematik ders kitabim idi
bu kadar yaygin kullanilan bir kitabın alistirmalarinin cozumu daha ulasilabilir olmali
bu kadar yaygin kullanilan bir kitabın alistirmalarinin cozumu daha ulasilabilir olmali
December 9, 2025
great for beginner mathematicians
April 18, 2021
The book is easy to read and explains every little detail very nicely. I enjoyed the structured and well-thought through presentation of the different proof techniques. Also I like that it discusses a bit of Calculus, which is usually not done in a discrete maths book (but the presentation is very basic; the connection to general proof techniques and the logic behind them is well presented though). In fact, I do not consider this book to be a discrete maths book. For this book to be a discrete mathematics book, it would need a structured treatment of discrete mathematics topics, amongst others some graph theory, etc. This certainly is a book about proof techniques and it presents them well, in parts better than other books.
Overall, I did enjoy reading the book very much.
It also is a great book for self-study since there are plenty of exercises with solutions which are great for self-testing.
Overall, I did enjoy reading the book very much.
It also is a great book for self-study since there are plenty of exercises with solutions which are great for self-testing.
December 23, 2025
The Holy Bible of Math.
September 1, 2019
A comprehensive and excellent introduction to proofs, includes answers to odd-number exercises.
Displaying 1 - 30 of 36 reviews