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Does anyone knows books which have lots of questions ,whose format are like fill in the holes type . . Same goes for theorems and exercises . I am looking on pure math especially Real analysis ,Abstract Algebra ,Topology etc

Thanks

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closed as too broad by Xander Henderson, Claude Leibovici, Jaroslaw Matlak, The Phenotype, Ethan Bolker Mar 26 '18 at 12:39

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  • $\begingroup$ There are nice books by R. P. Burn that do Number Theory, Group Theory in that style. $\endgroup$ – André Nicolas Jul 18 '15 at 17:22
  • $\begingroup$ @AndréNicolas Thanks $\endgroup$ – Sophie Clad Jul 18 '15 at 17:28
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You might be talking about "problem books" which contain very little exposition, and ask the reader to prove most of the propositions. For topology, you might try Elementary Topology, by Oleg Viro. Basically the structure is that of a guided inquiry from definitions through the major theorems. There are hints and answers in the book if you get stuck in places.

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  • $\begingroup$ I was about suggesting that one, you beat me to it. +1 $\endgroup$ – Ivo Terek Jul 18 '15 at 17:13
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    $\begingroup$ @IvoTerek Do you know of any others in this format? I really liked this book, and as a tutor/TA at my university, it would be really good to have this for other subjects - especially calculus since that's the subject I see the most students in. $\endgroup$ – Alfred Yerger Jul 18 '15 at 17:17
  • $\begingroup$ The only one that comes to mind now is John Oprea's Differential Geometry and its Applications, but I don't this subject interests the OP (at least for now). I guess I'll write an answer, just for completeness.. $\endgroup$ – Ivo Terek Jul 18 '15 at 17:25
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    $\begingroup$ No, thinking more, Oprea's book isn't like it. I don't know any other books in that exact same format. $\endgroup$ – Ivo Terek Jul 18 '15 at 17:49
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Although not being one of the subjects you asked for right now, John Oprea's Differential Geometry and Its Applications might fit the bill: there's lot of exercises in the middle of the text, eventually some details of proofs turn into exercises too, he doesn't have any section labeled "exercises", working through most of the examples given are exercises too, and nonetheless the book is very pleasant to read.


Halmos' Linear Algebra Problems Book is likely to suit your needs. He talks a bit, then throws you a problem. You'll interact a lot with the book.

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If in your "etc" you are willing to include some set theory, the two volumes by Just & Weese, Discovering Modern Set Theory, are a must-read in this vein.

I think it is relevant to add that, apart from being very well written, the first volume ("The Basics") is the only math textbook that made me laugh out loud!

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  • $\begingroup$ And actually, I've just read one comment by @SophieClad in SamM's answer that is the best description of these two books :-) $\endgroup$ – Pedro Sánchez Terraf Jul 19 '15 at 4:35
  • $\begingroup$ @SophieClad You're welcome. And you will certainly be pleased with this book, specially volume I. It also made me laugh out loud over some passages! I'll put this into my answer since it is another defining characteristic of the book. $\endgroup$ – Pedro Sánchez Terraf Jul 19 '15 at 13:16
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I've always been a fan of Rudin's books on analysis; Principles of Mathematica Analysis, Real and Complex Analysis. These books have numerous exercises, although the latter can be quite tough. Also worth looking at is Mathematical Analysis by Apostol. This book covers all the basics of mathematical analysis and has a lot of exercises.

For Topology, I recommend Sutherland's book Introduction to Metric and Topological Spaces, which is an excellent introduction to the topic and has plenty of exercises and is otherwise well-paced.

Abstract algebra is a little bit out of my expertise, but you might try A first course in abstract algebra by Fraleigh. If I remember correctly, this has plenty of exercises at a good level.

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  • $\begingroup$ I think OP wants books where a good number of the propositions in the book are in fact exercises for the reader, just guided by the exposition. $\endgroup$ – Alfred Yerger Jul 18 '15 at 17:16
  • $\begingroup$ Certainly Apostol and Sutherland have such propositions and, if I remember correctly, Fraleigh does too. $\endgroup$ – SamM Jul 18 '15 at 17:17
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    $\begingroup$ @AlfredYerger i think maybe yes, but i want books where say they give hint to start a proof ,then after thinking they give another step(which is not spoiler .but part of learning process) so that i can derive proof for myself $\endgroup$ – Sophie Clad Jul 18 '15 at 17:18
  • $\begingroup$ @SophieClad if you mean that they do this for every proposition, then these are the problem books I mentioned in my answers. While all the texts listed here are good, none of these problem books. $\endgroup$ – Alfred Yerger Jul 18 '15 at 17:19
  • $\begingroup$ In most books (in my experience), the proofs left to the reader follow very similar formats to other proofs presented by the author, some point this out explicitly. Plenty of textbooks present proofs which have details missing, which is probably what you are after. $\endgroup$ – SamM Jul 18 '15 at 17:22
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Abstract algebra by Allan Clark.

The book presents an average course in undergraduate algebra through a long list of questions , proves all the harder theorems but leaves almost everything else to the reader , with many hints and motivations.

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  • $\begingroup$ Would you elaborate a bit, i.e. why would you recommend this along the the spirit of the original question? $\endgroup$ – Frenzy Li Apr 7 '17 at 8:24

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