I have been self-studying mathematics for many years now and I've noticed that you don't really find mathematics books which have solutions to the exercises, and sometimes they don't have exercises at all. I think that if I could do exercises and compare my answers to the solutions, it would give me much stronger understanding of the subjects.
So I was wondering if there is a book that would contain problems from different subjects in mathematics with solutions. I know there are some such books but they're usually concentrated on only one or two subjects. It would be convenient if all the exercises were in one book.
I'm looking for a book in the undergraduate level which would cover as wide range of fields in pure mathematics. Areas I'm interested in are analysis, algebra, topology and number theory. I'm excluding fields such as geometry, elementary algebra and other fields that are a subject of competition mathematics (IMO) because there are already a lot of resources for those.
If such a book existed, it would be a very valuable resource for self-study. Thanks!
Edit:
This question was put on hold as too broad. The reason, why I think it's better not to restrict this question too much, is because I know there aren't many such books, that I've described above, if any. This way the people answering have more freedom to suggest anything close to what I'm looking for. And because there aren't many such books, there won't be a problem of having too many possible answers.
Also, when I don't restrict it too much, I think more people could benefit from the question and answers.
According to help center: "if your question - - has many valid answers (but no way to determine which - if any - are correct), then it is probably too broad" I don't think that this is the case with my question since the correct answer is the book with questions from as many fields as possible (and I did restrict it to fields mentioned above...).
Miksu, hello!
I'm not able to recommend a single book covering both algebra and analysis, at least in English. Also, from your question it appears you're at a level where you're still becoming accustomed to rigorous math, and what does or doesn't constitute a proof. Therefore the books I'm going to suggest are at a level generally somewhat below that of, say, Rudin's Principles of Mathematical Analysis.
The solutions manual to Spivak's Calculus. (The problem statements are in the textbook.)
Demidovich. Problems in Mathematical Analysis.
Halmos. Linear Algebra Problem Book.
Faddeev, Sominsky. Problems in Higher Algebra.
The last two books have a separate "Hints" section that comes before the answers.