Textbook recommendations for undergraduate proof-writing class 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:


*

*Logic, Sets, Equivalence Relations and Induction should be covered.

*Price should be reasonable (say around $30 or less).

*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!
 A: I have used Velleman's How to Prove It with success.
A: In addition to those mentioned, here is a good book which is just under $30:
Kevin Houston, How to Think Like a Mathematician: A Companion to Undergraduate Mathematics, 1st Edition 2009.
A: I quote from a recent article by Brown and Porter in the De Morgan Journal, published online by the LMS. http://education.lms.ac.uk/wp-content/uploads/2012/02/Brown_and_Porter.pdf 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 Then the various props are gradually withdrawn.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."
A: 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:
http://estymath.com/Proof.html.
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.
A: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?ie=UTF8&qid=1303491885&sr=8-1
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).
A: 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".
http://www.amazon.co.uk/Concise-Introduction-Pure-Mathematics/dp/1584881933
At least in the UK that book is pretty darn cheap.
A: I've been really happy with Smith, Eggen and St. Andre:
http://www.amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025/
Though that breaks your price requirement.
A: 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 


*

*Symbolic Logic

*Proof Methods

*Mathematical Induction

*Set Theory

*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 
http://www.jiblm.org/downloads/dlitem.aspx?id=17&category=mathnerdscollection
http://www.jiblm.org/downloads/dlitem.aspx?id=16&category=mathnerdscollection
http://www.jiblm.org/downloads/dlitem.aspx?id=14&category=mathnerdscollection
(These last three haven't been refereed by the journal, but they still gives links to them.)
A: The Book of Proof by Richard Hammack is free online and available from Amazon for $12.95.
A: 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).
A: Free online textbook on proof-writing:
Mathematical Reasoning: Writing and Proof by Ted Sundstrom
A: My father taught heat transfer in a mechanical engineering department and he had a great trick. Take a good text such as one suggested in another solution: 
http://www.pearsonhighered.com/educator/product/Mathematical-Proofs-A-Transition-to-Advanced-Mathematics/9780321390530.page
This book costs $100+
And then use an older edition for your class. You can buy the 1st edition of that textbook for $9 here:
http://a.co/1Qlm9sT
A: http://www.amazon.com/Proofs-Fundamentals-Course-Abstract-Mathematics/dp/0817641114
This is written by my professor Ethan Bloch. It is slightly overpriced, though. 
A: 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.
A: 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 http://www.dcproof.com
A: Jimmy Arnold has a full book available online (An Introduction to Mathematical Proofs):
http://www.math.vt.edu/people/elder/Math3034/
Also, Michael Hutchings has a very nice 27 page manuscript on the subject (Introduction to Mathematical Arguments)
http://www.math.berkeley.edu/~hutching/teach/proofs.pdf
A: 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.
A: Miklós Laczkovich: Conjecture and Proof
A: 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:
http://thoughtsseries.weebly.com/
A: 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:
Mathematical Proofs: A Transition to Advanced Mathematics by
Chartrand, Polimeni, and Zhang
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 sophistication ... to my taste, of course.
A: 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.
A: 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:
http://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/0471406473/ref=ntt_at_ep_dpi_3
http://product.half.ebay.com/How-to-Read-and-Do-Proofs-by-Daniel-Solow-2001-Paperback/948996&tg=info
A: I'll throw in this book, as well, since it's what I used with pretty good success:
http://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094
