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I'm trying to increase my skill with proofs and at least one of my bottlenecks is skill with set theory proofs. What are good largish sources of practice problems for set theory proofs (with answers if possible)?

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Your question is vague because lots of people inquiring about this mean the sort of thing taught under the heading of "set theory" in very elementary courses, where you do algebra with unions, intersections, complements, etc., whereas others have in mind things like transfinite ordinals, etc. What level do you have in mind? –  Michael Hardy Oct 7 '11 at 22:30
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5 Answers

there are various sample proof pdf online you can use to get yourself better antiquated with what style of proof to use. this might help

http://www.math.sc.edu/~cooper/math574f10/problems2.pdf

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I hope you will like this book also: Problems and Theorems in Classical Set Theory (Problem Books in Mathematics)

by: Péter Komjáth, Vilmos Totik,

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Here is the website of the book (on the webpage of one of the authors): cs.elte.hu/~kope/setproblems.html and here's the google books link: books.google.com/books?id=rbCmt-2NxtIC –  Martin Sleziak Nov 29 '11 at 14:25
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I'm not sure of the level you are interested in, but you could try Schaum's Outline of Set Theory. The entire "Schaum's" approach is to motivate concepts by working problems.

If the cost is prohibitive, you could search Google Books with phrases like "elementary set theory", "introduction to set theory", etc. With luck, you might stumble across a full version of some other text.

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You could try some set theoretic books:

  1. Thomas Jech, Set Theory (3rd Millennium ed.);

  2. Paul Halmos, Naive Set Theory;

  3. Azriel Levy, Basic Set Theory;

  4. Herbert Enderton, Elements of Set Theory.

There are many many more books. Each book has many questions, you can also try and prove some of the lemmas and theorem by yourself and then compare with the given proof appearing in the book.

The important thing is not to cheat yourself, if you're writing a proof and you're not 100% convinced in something that you have written - write it again. If you are not able to write it formally - write it again.

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Halmos's book is great, but contains few problems. A nice "companion book" with lots of problems (at that level) is Exercises in Set Theory, by L.E. Sigler. Unfortunately, it is hard to find these days. My copy (actually, my dad's copy) was published in 1966 by Van Nostrand. Cheapest prize I found on BookFinder was about $70. –  Arturo Magidin Oct 7 '11 at 19:43
    
@Arturo: I personally had exercises from courses, and coming up with my own ideas for exercises (supported, of course, by two wonderful teachers I had). By the time I met Jech's book I was reading the advanced part. I do know that the introductory course given in my university (which I TA during fall semesters) is based on Levy's book for the most of it. From there to the books here it's all hearsay. Nonetheless, for formal practice pretty much every math book should be enough. –  Asaf Karagila Oct 7 '11 at 21:03
    
I wasn't trying to correct you; rather, I was trying to add a bit to your answer... Sorry if it came off the wrong way. –  Arturo Magidin Oct 7 '11 at 21:19
    
@Arturo: I apologize, then. It seems my word came out in a defensive tone, while I did not intend this. Many thanks for the comment! :-) –  Asaf Karagila Oct 7 '11 at 21:51
    
A very good book at the advanced undergraduate/first year graduate level is K. Hrbacek & T. Jech, Introduction to Set Theory. –  Brian M. Scott Oct 9 '11 at 6:30
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Halmos' Naive Set Theory contains a few problems, though some of them are a bit difficult.

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