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I would like an up-to date book which lists results with perhaps a comment on how to prove it, as this one. I have gone through the algebra section partially and I am awed. It provides definitions, lists results and has occasional comments. I try to prove, make my own examples and connections and discover it as much myself before referring formal books. I would like to know if there are similar books which are updated and include more advanced areas of mathematics as well.

EDIT1: Thanks for the answers. I plan to give the other answerers bounties as well once I have higher reputation.

Edit2: Related question on MO "Books with a theorems as problems approach"

Edit3: Sorry this question keeps appearing on top. Please refer: meta post

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When you read an author's definition, you have inevitably already started absorbing some part of their way of reasoning: making definitions is an important part of the mathematical process. In any case, your rhetoric seems a bit unfair to say the least to most textbook authors. I hardly think they are trying to force anyone to do anything. –  Qiaochu Yuan May 1 '11 at 5:24
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I agree with Qiaochu about your second paragraph, but +1 for the interesting question in the first. BTW, for those not in the US (Google doesn't show the book even though it's from 1880 and in the public-domain), the book is here on archive.org or here. –  ShreevatsaR May 1 '11 at 5:35
    
So is there any updated equivalent. –  kuch nahi May 1 '11 at 15:15

4 Answers 4

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Not quite the same thing, but there are books for self-learning that are similar even if much restriced in scope. See for instance, A Pathway Into Number Theory.

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For Linear Algebra, Paul Halmos has written "The Linear Algebra Problem book" that comes pretty close to what you want. For topology, take a look at "Elementary Topology Problem Texbook" by Viro, et. al.

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Somewhat of a different flavor I believe (I did not read Carr), but I think the princeton companion to mathematics might be something you are interested in. It is one of my favorite books in mathematics. It has some long articles introducing different parts of mathematics in depth, with current results and aims for the future. Also many theorems are explained in laymens terms (a mathematical laymen, i.e. not an expert in that particular field) in an encyclopedian part of the book.

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For Combinatorics you might want to take a look at this book, whose author is also quite well known.

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"This book" is Combinatorial Problems and Exercises by Laszlo Lovasz. –  ShreevatsaR May 11 '11 at 9:01

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