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I am a biochemistry graduate student who works on cancer. I am interested in learning proofs as a personal interest. I use math as a tool, but would like to start building a deeper understanding on my own. I am not taking any course. Hence, I am looking for a book with theory, exercise, and solution manual, in case I am stuck. I find this forum extremely helpful, but I would still like to have a reference. Most books that I have started looking to buy do not have a solution manual. Can anyone recommend an author? Sorry for this general question. Thank you!

EDIT:

I watched the movie Good Will Hunting, so I feel confident! lol I

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  • $\begingroup$ How to Prove It: A Structured Approach by D. J. Velleman. The book has a lot of hints and solutions. Lots of the problems can also be found here on MSE. You can also find the solutions here. $\endgroup$ – Git Gud May 23 '15 at 23:19
  • $\begingroup$ what would be the point of it? it is not like there is only one proof. $\endgroup$ – abel May 23 '15 at 23:21
  • $\begingroup$ I don't think many mathematicians ever read a book just on how to do proofs, you just start reading math books and kind of get the idea of proofs as you go. I suggest just start with a good book on either abstract algebra or real analysis and just start reading from chapter 1 and see if you don't just pick up how it works. $\endgroup$ – Gregory Grant May 23 '15 at 23:30
  • $\begingroup$ @GregoryGrant If you think someone who's never done rigorous proof-based math can just immediately learn abstract algebra or real analysis (especially without a teacher!) you might be a little farther removed from the non-mathematical world than you think. These courses are often intended to be taught to third or fourth year pure math majors at universities, i.e., people who already have experience with proof techniques. $\endgroup$ – Samuel Yusim May 23 '15 at 23:35
  • $\begingroup$ @SamuelYusim Maybe in the U.S., in Italy these are first semester first year courses. Are Americans really that much worse prepared than Italians? $\endgroup$ – Gregory Grant May 23 '15 at 23:37
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In my experience you won't tend to find a book with a solution manual and if you do you won't find one that does a lot of exercises as proofs can get pretty long and tedious in a hurry. Most authors, I think, assume that you will be interacting a fair amount with your professors/other students at that point so I don't think they get written very much.

For a basic text into abstract math I would suggest my undergraduate text "Bridge to Abstract Mathematics" it does a good job of making important remarks and breaking things down. There is, no surprise, not a solution manual that I am aware of but I have a set of YouTube videos going up that go through and lecture on each chapter and do a number of the examples/problems in detail. They will start going up in a few weeks under the username superphyz and I will link them in the comments later if you comment and let me me know if that will be useful to you.

Good luck!

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  • $\begingroup$ Thanks! That sounds really good! I have a learned a great deal from YouTube videos thanks to helpful people! :) $\endgroup$ – Johnathan May 24 '15 at 0:03
  • $\begingroup$ No problem. I will let you know as soon as they start going up. I'm writing most of them up now so I can get them up in a fairly timely manner. So one lecture plus the related examples should go up once or twice a week. $\endgroup$ – Aaron Zolotor May 24 '15 at 3:13
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Taken together, Ian Stewart and David Tall's 'foundations of mathematics' (a really well written and super readable way into modern maths) and Martin Liebeck's 'concise introduction to pure mathematics' (3rd edition has solutions, often used in the UK for students preparing for Oxford/Cambridge) would constitute a good start. I teach students bridging from 16-18 yrs to university and use these books a lot, they both offer a lot to interested readers without being overwhelmingly abstract and terse.

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  • $\begingroup$ Hi! Thank you for your reply. Is the solution manual separate? $\endgroup$ – Johnathan May 25 '15 at 21:40
  • $\begingroup$ Liebeck contains solutions to odd-numbered problems within the book itself, but only in the 3rd edition, so watch out for that. Stewart and Tall does not have solutions, but is such a good book I have to recommend it, even though it's not quite what you are after! $\endgroup$ – AmpleMimic May 25 '15 at 22:37
  • $\begingroup$ @AmpleMimic I intend on sitting the STEP exams in the future, do you know of any books that you can recommend for them? $\endgroup$ – seeker Sep 11 '15 at 8:35
  • $\begingroup$ @seeker Liebeck was a recommended text for Oxbridge candidates in my school days, but my favourite resource (and free, and available at the click of a button!) is Cambridge University's Stephen Siklos' booklets, which can be found along with a lot of other useful material here: mathshelper.co.uk/oxb.htm $\endgroup$ – AmpleMimic Sep 11 '15 at 8:57
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Here is the book I used in my proofs class. It is a free online PDF, and the book can be bought for under $20. In the solutions sections of the book these are many written out proofs. Going to lectures does help but honestly, I am learning from the book.

Best of wishes!

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For a non-mathematician I would recommend the following books

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Well, there are lots and lots of books with solutions.

An answer to your question depends on which part of mathematics you would like to explore. For instance, a series of great ones about real analysis come from Kaczor & Nowak

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