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I am teaching the proof-writing class (for the 3rd time) in the Fall and plan to buck the party line and use a different text than the default Bond and Keane. My parameters are as follows:

  1. Logic, Sets, Equivalence Relations and Induction should be covered.
  2. Price should be reasonable (say around $30 or less).
  3. Distractions like "historical comments" and "mathematical perspectives" should be kept to a minimum.

I plan to supplement such a book with "What is Mathematics" by Courant and Robbins.

I would be pleased to hear some recommendations!

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In this for math students? The fact that there is a proof-writing class makes me sad... That we have managed to dissociate so much proofs from learning calculus and algebra that a separate course is needed is quite a feat in absurdity! – Mariano Suárez-Alvarez Apr 22 '11 at 17:32
(There was a nice letter to the editor by Maclane on the AMS Notices. where he argued that the reason to teach calculus is to teach logic (I guess he included proof ẃriting in that...), something like "one shoul dteach enough calculus so that the logic comes across". I cannot locate it) – Mariano Suárez-Alvarez Apr 22 '11 at 17:51
@Mariano: unfortunately, we have to take the world as it is even while trying to make into what it should be. In other words, I don't think it's fair to suggest as you do that the need for a proofs course stems from a screw-up in curriculum design. I never had a "proofs" course, never taught one, and I don't relish the idea either. But for those of us who have calculus classes that mix math majors with people who intend from the beginning to stop at Calc 1 (pre-med anyone?), there is only so much we can do in those classes and the case for a proofs course more or less makes itself. – Thierry Zell Apr 22 '11 at 20:33
I have yet to understand why abstraction is commonly taught at the same time as rigorous proof-writing. They're the two most important skills for undergraduates to learn, and they're different skills. IMHO, combinatorics is an excellent subject for learning to write rigorous proofs, precisely because the definitions are easy to understand, and you don't have to spend a lot of time proving theorems which "look obvious". – Frank Thorne Apr 23 '11 at 6:20
@Frank: I find your comment intriguing, the more so because I can't decide whether I agree with it or not. ("Having an opinion" is not usually a problem for me!) If you're right, then we math pedagogues of the world are missing out on something rather important and fundamental. I encourage you to think and say more about this -- maybe via a MO question, maybe via email. – Pete L. Clark Apr 23 '11 at 19:06

21 Answers 21

If you want a book which is priced under $30, write it yourself and put it on the internet: then it's free.

(This is not a quip or a dismissive comment: please do actually do this. I have done this sort of thing myself.)

Among books that the evil empires of publishing put out, I used one for such a course twice and -- apart from the price -- it was pretty good:

I'm not sure exactly why you are against historical comments (nor do I know exactly what "mathematical perspectives" means in this pejorative context), but so far as I recall this book is fairly businesslike. (Added: I just processed the part of your question where you mention supplementing the book with material from Courant and Robbins. That latter book is all about perspective, so I guess the idea is that you want to avoid duplication of content, which is very reasonable. Sorry if I sounded overly critical before.)

I was most pleased with the treatment of logic and sets in the first two chapters: just about the right amount, with just about the right level of formality and my taste, of course.

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I second that idea,Pete. – The Mathemagician Apr 22 '11 at 18:37
Our university has used the Pearson book now for a while, and it seems to garner general acceptance. I like it, although there were a few oddities here and there. Overall a nice book. – Pace Nielsen Apr 22 '11 at 19:15
I just might do this after I teach it a couple more times (and get tenure!) – Eric Rowell Apr 23 '11 at 16:44
An alternative to giving it away is to use print on demand (POD) publishers. I have been very happy with Createspace, who give a good royalty rate, as they are an amazon company, and are very efficient. Of course you have to do the publicity yourself. I discuss this on although some details are out of date, since Booksurge has become Createspace. – Ronnie Brown May 15 '12 at 20:53

I learned out of Dan Solow's How to Read and Do Proofs and it was great (this was about six years ago and the professor had used this book for many many years). It's also very cheap:

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I recommend anything but this book. When I taught out of it, prepared students liked it a lot, but borderline students (who need such a class more) struggled more than they might have without any book. It hurts weak students. It seems to feed an expectation in them that all proof-writing is done "step-by-step", with the precise sequence of steps dictated entirely by the formal structure of the statement to be proved, and the exercises do not carefully delineate between what math can be taken for granted and what cannot--- giving the impression that "proof writing" can be done in a vacuum. – anon Apr 26 '11 at 0:17
Hmm, I guess I can't comment about how the book is for "weak students" but I will say that when I took this course I had no experience with proof writing whatsoever. I was pretty hopeless when I started, but the instructor met with me many many times and this book was a great supplement for those meetings. In particular, I learned a ton from the "backwards-forwards method" of solving a problem...reducing your problem to simpler pieces and solving those. I can't remember well enough to comment on the exercises, but I do know the book covers all proof methods you see in early undergrad math – David White Apr 26 '11 at 14:27

This year, my colleague has been using the art of proof by Matthias Beck and Ross Geoghegan (Springer 2010). It's slightly below $40 I believe, which is still in the reasonable range, commendably short and I hear it's proved very satisfactory so far. I think it has the topics you're looking for.

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This book is also one of the ones springer has already uploaded to their "SpringerLink" website, so some universities might even have a subscription making it freely available to university IPs. Here's a link: – Rob Harron Apr 25 '11 at 15:31

In addition to those mentioned, here is a good book which is just under $30:

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I like this book very much, but I wonder if it has too many "distractions" for the questioner. – Carl Mummert Apr 25 '11 at 17:15

I quote from a recent article by Brown and Porter in the De Morgan Journal, published online by the LMS. and commented on subsequently by David Wells. I feel the following idea needs advertising.

"A technique widely used by psychologists and trainers is error-less learning. This falls into two types. One is where large hints, props, and supports to a specific course of action are given, and the action is rewarded as a symbol of success. Then the various props are gradually withdrawn. The other type uses reverse chaining: the easiest way to see to this is to think of encouraging a child to put on a vest. You do not throw him or her a vest and say put it on; instead, you put it almost on, and then ask the child to do the final action. Subsequently, you gradually put the vest less and less fully on, till the whole action can be done.

One way of using the last technique in university mathematics is to write out a formal proof and then erase bits of it. The student has to fill in the bits, using clues from the rest of the proof. This has some analogies with the practice of a professional mathematician, who may have an idea and outline for a proof, but needs to work on details. The student also gets an idea of the structure of a proof. Such an exercise is also very easy to mark!

The general feeling about error-less learning is that it works like a dream!

In either method, the fact long verified by psychologists is used, that we learn from success. We can also learn to accommodate failure if that is gradually introduced, and strategies are available for dealing with failure."

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But in Karate Kid (the 2010 version), Jackie Chan trains the kid by asking him to put his vest off, then hang it on, then put it on, then put it off, etc., and at the end of the movie, he IS the champion... Is math so different from kung-fu? – ACL Jun 30 '13 at 23:15
@ACL: Having taken up Tai Chi in my old age, and noting that Tai Chi was derived from martial arts by some who got muscle bound, I am inclined to think that maths teaching can learn from the contrast! My Tai Chi class has all sorts, including some quite fit, and others coming with sticks and even zimmer frames. The class gives 1.5 hours of gentle exercises which one tries to do with a sense of rhythm, and with no competition. – Ronnie Brown Jul 1 '13 at 9:16
@ACL: Re Karate Kid: I tend to think that film was not reportage, but fiction! It may be relevant, or not. The methods of errorless learning are used by animal trainers and with children as a matter of course. Another technique is to start with so many "props" that success is achieved and then gradually remove the props. It goes with trick in Polya's book on problem solving: start with the simplest possible case. – Ronnie Brown Jul 1 '13 at 9:29
My comment was prompted by the mention of the vest, and was not to be taken too seriously. – ACL Jul 2 '13 at 10:57
Surprisingly, when it comes about sports, selection and competition are considered as essential. I practice yoga myself, and I agree that I would love to see/make a math class driven like a yoga class. No competition, everybody has to achieve its personal goal. In comparison, what we/I propose is closer to ranking, selection, elimination, etc. But amazingly, our students to not feel they are voluntarily engaged in the learning of mathematics; what about yours? – ACL Jul 2 '13 at 10:59

I have used Velleman's How to Prove It with success.

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I use "Proof: An Introduction to Higher Mathematics," by Esty & Esty (my father and me). We self-publish, in order to keep it relatively cheap--I think bookstores sell it for about $45, depending on the markup. The chapters are split into two categories, theory and practice:

(Theory) 1: Intro to proofs (sets, logic) 2: Sentences with variables (generalizations, existence, negations) 3: Proofs (inequalities, absolute values, contradiction, contrapositive, induction)

(Practice) 4: Sets (set theory, bounds) 5: Functions (one-to-one, onto, functions on sets, cardinality) 6: Number Theory 7: Group Theory 8: Topology 9: Calculus

At the liberal arts college where I teach, we generally get through the first five chapters (in a one-semester course). One could skip around more than I do, however.

The main thing our book does differently than others is emphasize a lot of common grammatical mistakes students make when first learning proofs. We found a lot of proof books already assumed that students understood a lot about the language we use when we write proofs, and only taught specific techniques like induction. We spend more time on the language at first, including conventions as well as logic. Probably for strong students who already possess good mathematical intuition it would be unnecessary, but we've found it works better for our students. The book has been used at Montana State, Marshall, Case Western, Boise State, Texas State San Marcos, etc.

If you think you're interested, there's more info here:

It seems some faculty want proofs in combinatorics and equivalence classes. Our thoughts were that we wanted to prepare students for classes with many definitions of terms and proofs using them, such as Advanced Calculus, Real Analysis, Linear Algebra, or Abstract Algebra. Combinatorics has a method of proof all its own that is not seen much in those classes, so we omitted it, and we do only a bit of equivalence classes because they are short and easy given what we cover. So, if you need a lot of combinatorics, our book is not the right one for you. If you are at a high-powered school with very strong students our book is not the right one for you. However, if your students make the same sort of logical and grammatical mistakes commonly seen in "Introduction to Real Analysis," this text may be right for you. 

I teach at Stonehill College, where we have a proofs course called "The Language of Mathematics," which is taken by math majors after Calc II, concurrently with Calc III. We introduced the course a few years back because we found that students weren't really prepared for the rigors of analysis or algebra, and that a lot of time was being spent in all upper division courses teaching the same stuff. Things are definitely better since we've added the course.

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Hi Norah. Nice seeing you here. – Andrés Caicedo Jul 12 '13 at 17:59
(I've used this book myself, and convinced a few colleagues at Boise State to use it as well.) – Andrés Caicedo Jul 12 '13 at 17:59
I also see the notion of proof as part of the Methodology of Mathematics, and which needs discussing with students. See this article: – Ronnie Brown Jan 19 at 22:23

This text was used in the "Math Structures" class at my undergraduate institution (basically an intro to proof writing) and I found it really useful for transitioning from calculus type problems to constructing proofs. I think it meets all your requirements (definitely the first two, and I don't recall there being a great deal of historical\philosophical digressions).

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I second this: Velleman's book "How to prove it" is quite usable and not too expensive. I would say it focuses quite a lot on the mechanics of how to attempt a proof (how to prove a statement of the form if A or B holds then C holds). For strong students this is probably unnecessary, but for average students it's very useful. – Anthony Quas Apr 22 '11 at 20:35

I am not so sure of the US system but one of the books we recommend at our university is Martin Liebeck's "A concise introduction to pure mathematics".

At least in the UK that book is pretty darn cheap.

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I should add the "disclaimer" that in fact Martin Liebeck works at my university! I believe he wrote the book because he couldn't find "the book that he wanted", but this was before the days when one could self-publish so he couldn't follow Pete's advice... – Kevin Buzzard Apr 22 '11 at 20:32

A "book" that satisfies all of your criteria is a set of notes from the Journal of Inquiry Based Mathematics called "Introduction to Proof" by Ron Taylor. linky

The chapters are

  1. Symbolic Logic
  2. Proof Methods
  3. Mathematical Induction
  4. Set Theory
  5. Functions and Relations

There are two appendices: one on mathematical writing and one on Style (By James Munkres).

It is a set of notes for an IBL class, so the assumption is that the students will be doing virtually all of the proofs themselves. I've never used this set of notes for teaching, but I've used others from the journal. I like them very much.

Their copyright notice allows free use and printing as long as attribution is given and no charge for the students other than printing costs. Similar sets of notes that I've used have cost the students about $6.

Others from the journal's website about intro to proof/foundations are

(These last three haven't been refereed by the journal, but they still gives links to them.)

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Thanks! I really like Taylor's notes. I will definitely include this as an extra resource if only for the advice in the appendices. – Eric Rowell Apr 26 '11 at 15:34
It is interesting to see that the notions of category theory are not mentioned in these answers. In particular, the proof by verification of or use of a universal property is very powerful. Many examples of this are in my book "Topology and Groupoids": for example in general topology (products, sums, quotients); in connection with the Seifert-van Kampen theorem; and in connection with the fundamental groupoid of an orbit space. – Ronnie Brown Jul 1 '13 at 9:38

Jimmy Arnold has a full book available online (An Introduction to Mathematical Proofs):

Also, Michael Hutchings has a very nice 27 page manuscript on the subject (Introduction to Mathematical Arguments)

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I've been really happy with Smith, Eggen and St. Andre:

Though that breaks your price requirement.

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This is written by my professor Ethan Bloch. It is slightly overpriced, though.

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Still priced a lot more reasonably than the book by Douglas Smith et al. – Deane Yang Apr 22 '11 at 17:07
I am aware of that. But I think you can get a personal discount from him by 25$ per book if you used it for your class, which is the cost we used to buy it. – Kerry Apr 22 '11 at 22:22

Peter Eccle's "Introduction to Mathematical Reasoning: numbers, sets and functions" seems to fit the bill of what you are looking for. It is slightly higher than your preferred price of 30 dollars (it is 38). I would also check out the Google books preview.

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I have taught out of this book and second the recommendation. (But yeah, the price.) – anon Apr 26 '11 at 0:01

The Book of Proof by Richard Hammack is free online and available from Amazon for $12.95.

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Miklós Laczkovich: Conjecture and Proof

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I think this book would be great for somewhat more capable students, but not for the ones likely to take the course in the question. – Andreas Blass Jun 30 '13 at 23:59

You might try Stoll's "Set Theory and Logic." It used to be available from Dover. I would assume the price would still be reasonable. The book does not have a specific section on proof techniques or strategies. However, I have always preferred to discuss these myself with my own examples, usually from set theory in the beginning. Having technique and strategy material in a text always struck me as trying to make math too formulaic. This is probably just a quirk of mine (but I also believe in full disclosure, within reason).

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There is an online free book called Thoughts - alpha this book is a compilation of mathematical proofs for basic mathematics (Trigonometric Identities, logarithms, volumes and surfaces, basic series and basic calculus) it might be helpful!

Thoughts - alpha webpage:

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Not a book, but it's free. May I humbly suggest my DC Proof software. Using a very user-friendly proof-checker, students can work through a ten-part tutorial that introduces various methods of proof. For more information, free download, testimonials, etc. visit my website at

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The book I used in my 'proofs' class was "Doing Mathematics: An Introduction to Proofs and Problem Solving" by Steven Galovich, here on Amazon.

The class was called "Mathematical Structures", which is an apt name since the class wasn't solely about learning to prove things. It was learning to prove things in the context of learning about basic mathematical objects. It starts with basic logic, but after it introduces sets, relations, functions, equivlance relations and the like, it goes onto to develop the ideas of cardinality, including Cantor-Bernstein. It also has a couple other topics, like some basic combinatorics, the constructions of number systems, or looking at consequences of the field axioms.

It was a great introduction to what math is "really about" coming after some mostly computational calculus and linear algebra courses. The price is about $50, so it is a little more than you were looking for. But it is absolutely a book worth having.

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