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The problem sets that you usually get in a university course is a small fraction of the exercises in your textbook. Which raises a question: do you need to solve all the exercises from your textbook? Maybe the author just presents a lot of them so that your professor can choose those that fit his course.

I'm looking for a reason not to do all the exercises, or even only do the course's problem sets (a small fraction). For example, Pugh's "Analysis" claims to have more than 500 exercises—that's way to much; I think the bulk of them is not essential. The main idea of studying mathematics is to learn new mathematical concepts, and not get bogged down in routine computations.

What is the reason of the problem sets? Maybe just to check that you understand the concepts well. But maybe the problem sets are so small because they are meant to be checked by some person, otherwise they would be bigger.

EDIT: Especially it relates to the courses that you need as a prerequisite for the other much more important course. For instance, I need Linear Algebra as a prerequisite for Analysis. When I get to Analysis, I will do all the exercises. But I don't want to spend a lot of time on Linear Algebra—I understand the concepts, understand the proofs, and that's it.

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I think this question can also be asked at MESE. –  Git Gud Apr 4 at 19:14
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What I like to do is glance through the problems and see if I know how to do them. If I already have done a similar problem, then I don't bother, otherwise I invest time in doing it. –  Grid Apr 4 at 19:15
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Two exercises that I skipped: "Pick up any book on homological algebra and prove all the theorems without looking at the proofs in the book" (Serge Lang's Algebra), and "Do any fifty problems in Kelley's book" (Reed and Simon about John Kelley's Topology). –  Per Manne Apr 4 at 19:59
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@PerManne: Knuth's "Art of Programming" has an exercise where you're asked to prove the Last Fermat Theorem. The book had this exercise even when the theorem was a research problem. –  Graduate Apr 4 at 20:07
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@Graduate I'd probably skip that one too... –  Per Manne Apr 4 at 20:11

6 Answers 6

up vote 7 down vote accepted

I would say it somewhat depends on the level, but may be not that much actually. The following is how I did it myself when mostly in self study mode and no professor to oversee the progress.

As an undergrad, say in vector analysis, I did all the theory problems and enough many of the computational exercise to feel confident that I can do the rest. At that stage if upon reading another problem I could see a way to do it, then I wouldn't bother, unless the problem had some intrinsic appeal.

As a beginning grad student it was more or less the same way, but as the exercises were largely theoretical I ended up doing most of them (they were fun actually). Later on it depended. If I only needed to get a general idea of the material, or felt eager to get to the next chapter, I would only a few exercises, and try to move on. If I skipped too many of the exercises, I would start feeling rather lost a few chapters further down. Then it was time to try the problems in the preceding chapter. If I couldn't, then I would go back to the preceding chapter, and so forth. Doing this iteratively worked quite well for me.

Of course some more advanced textbooks don't have exercises. Then you need to make them up yourself and otherwise apply whatever habits have worked for you in the past.

The preceding paragraph is kinda my main point. You need to find the way that works for you. Lower level textbooks offer more repetitive work, and you can cut some of that. But at your own peril!

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"If I skipped too many of the exercises, I would start feeling rather lost a few chapters further down. " In fact, some authors lodge significant concepts or problem-solving approaches within the exercises. So skipping some problems at one point may mean missing something they use later in the book's presentation. –  RecklessReckoner Apr 4 at 20:41
    
So, you kept track of those exercises that you did, maybe by marking them with a pencil? –  Graduate Apr 5 at 0:09
    
@Graduate: Correct. I circled the ones I had done. –  Jyrki Lahtonen Apr 5 at 4:48

You don't necessarily need to do all problems in your textbook, but you need to make sure that you can do them. This usually involves doing a reasonable sample to test yourself.

The exercises I explicitly give my students in their assignments are a minimal sample, and I always make it clear that they are not enough to master the subject. I also tell them what it means "to be able to do an exercise": it means to be able to do it without help, without looking at the textbook, in a reasonable amount of time, and correctly. I have learned from extensive experience that the last sentence is not obvious to a lot of university students.

Of course, the more advanced the course the less the previous paragraph applies. For more sophisticated subjects the exercises tend to be more complicated, and not just a direct application of the topics considered.

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Your last point is true.Indeed,this is why most serious graduate mathematics courses are essentially problem courses with little discussion. At that level,students basically need to build most of the theory themselves or the entire enterprise is useless. –  Mathemagician1234 Apr 4 at 19:22
    
@Martin Argerami Martin, I think you know that I try to read the book "C*-algebras and Finite-Dimensional Approximations" by myself. Do you think the exercises in that book are important? And during reading the book, I meet with many problems which I could not solve immediately, how to deal with these problems? Should I solve them one by one or ignore some of them? (Fortunately, some of problems are solved in this forum) –  Yan kai Oct 12 at 16:13
    
In that book in particular, many exercises cover essential parts of the theory or at least examples of it, so I think they are very important. It is also a book that requires a nontrivial background in functional analysis and operator algebras. –  Martin Argerami Oct 12 at 20:07

Uh,you paid for all of them,so why not at least try them all? Or as many as time allows.

I'm only partially being sarcastic here. We learn mathematics by doing mathematics.This is particularly true of analysis, where the concepts and methods are creative and require some ingenuity in attack. And that means developing experience with solving many different kinds of exercises, from routine computations to difficult proofs. You can read 100 books from cover to cover and have total recall-and I can garuntee you won't be able to do more then pass a standard exam without working at least some of the exercises.

My advice-do as many exercises as time allows. Also-if the exercises are asking you to just do tedious computations and/or restate definitions-then chances are you're not using the right book. If an exercise doesn't make you think about the question for at least a minute before you begin attempting to solve it,then it's going to be a useless exercise to do. Period.

And in closing-Pugh's book has a truly outstanding collection of exercises and I'd strongly advise you try as many of them as you have time for.

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+1 for the comment about Pugh's book. Rather than being a bunch of repetitive and routine problems, pretty much every single one of them illustrates an important concept which you should know if you want to be able to say that you've mastered analysis. –  Santiago Canez Apr 4 at 19:50
    
+1 by Pugh's book this and Stromberg's are the best book in Real Analysis that I've seen so far. –  Jose Antonio Apr 4 at 21:39

You should do as many exercises as you need to, and you should have sufficient self-awareness in relation to the subject to distinguish between "need" and "want".

Exercises are not an end in themselves. They are a means of learning the subject. If you understand a topic after solving a small number of exercises you can certainly skip the rest - perhaps come back to them later for revision. If you see that you can do an exercise without writing it down, then don't write it down. (But a word of warning: it's easy to fool yourself on this point. Maybe you should write it just in case.) On the other hand, if you finish all the exercises and still don't feel that you understand the topic, seek out some more. In some cases you will be able to repeat the exercises you have already done, changing the numbers to make them slightly different; in other cases you will need to ask your instructor for suggestions. If you finish a ridiculously large number of problems and still don't fully understand the topic, it may well be that your difficulty is not with the topic itself but with something in the background. In this case you should certainly seek advice from your instructor.

One final comment: once you have mastered a topic, you will still want to do the occasional exercise so as to keep yourself "in form". Think of the world's best athletes - once they're on top, do they stop training? Of course not - they keep it up so that they stay on top. Mathematics is not really very different from that.

Good luck!

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(I've put my remarks on book exercises in a comment above.) Something I tell students is that the point of practicing the problems is to find out what you don't know how to do. (Better there than sitting at an exam -- or needing something in your own work later on...) Your life-work is not solving all the problems in a book, but a decently-prepared text is intended to guide you in what is important to understand in your work (theoretical, applied, whatever) with the ideas discussed. So it will be important to you to be thorough, but you will never have time to spend on everything in one book (unless -- maybe -- it's fairly brief).

I don't want to repeat too much of what everyone else has already said. You won't master everything during the time of a single course anyway. The important thing in mastery is steady work, coming back to the ideas (and problems) repeatedly over time. Really great books are ones you find you can still learn things from later, as your own experience grows. This is why some books are worth keeping for a lifetime.

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Helpful Tip: I like to use a random number generator, from $1$ to the number of exercises in the section, to get $5$ or $10$ questions from the section exercises, then pretend I'm taking a test. You can do this multiple times, of course with new sets of numbers. Most of the time this will give you a broad selection of questions and will replicate the randomness of tests.

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For some books, you may need to restrict the sets in the section exercises you choose randomly from. Many of the higher-numbered problems would be terrible to put on an exam: they would require either too much "writing", or too much deep analysis, to have to work on "cold" (particularly under a time limit). –  RecklessReckoner Apr 4 at 22:11

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