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Real analysis before set theory?

  1. Jul 30, 2009 #1
    Hey guys I signed up for a "set theory and topology" class for the fall and was planning on taking real analysis in the spring. Set theory got canceled and so i am taking real analysis instead and pushing set theory to the spring. Is this a wise idea? Taking real analysis before a formal set theory/metric space class? Here is the curriculum for each class.

    set theory: http://courses.illinois.edu/cis/2009/fall/catalog/MATH/432.html?skinId=2169"
    real analysis: http://courses.illinois.edu/cis/2009/fall/catalog/MATH/447.html?skinId=2169"

    If this is not such a good idea, then what can I do to prepare? Thanks guys.
    Last edited by a moderator: Apr 24, 2017
  2. jcsd
  3. Jul 30, 2009 #2


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    This is undergraduate real analysis, so you don't need the set theory. Your real analysis course covers countability (from set theory) anyway and you won't need ordinals for undergraduate stuff. The axiom of choice starts to come in when you start constructing non-measurable sets, which belongs to measure theory covered in graduate real analysis. The topology class is going to use it for stuff like Tychonoff's theorem (for arbitrary products), which is why it's covering set theory, but that doesn't really have applications in undergraduate real analysis and won't be needed there.

    Of course it is good to know stuff like countability in advance. It helps you understand easy proofs like the fact that a monotonic function has only countable many points of discontinuity etc. However, the significance of this doesn't really show up before you start using the Lebesgue integral which is in a graduate level class.

    Summary: don't worry.
  4. Jul 30, 2009 #3
    But doesn't real analysis heavily use metric spaces? And what kind of counting do I need to study? What website/book can I look at?
  5. Jul 30, 2009 #4
    If you don't know set theory, I'm guessing, that you won't know how to do proofs? Or do you? If you do, I think you'll be fine (with lots of hard work), other wise you should learn how to do proofs before you get to the course, for which I recommend Vellerman's book.
  6. Jul 30, 2009 #5
    Yeah, I know how to mathematical proof with quantifiers, implications, negations, and all that stuff but I dunno I guess I'm just worried.. I don't know infinite set theory or deeper set theory or basic topology of metric spaces.
  7. Jul 30, 2009 #6


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    Well this infinite set theory that you seem to be afraid of is just the concept of cardinality. This is just one of the things that's covered on any course in real analysis, so I wouldn't be worried. Of course, it would help if you knew the stuff in advance, but that would apply to just about anything on the course. However, cardinality is probably one of the easiest things on that course.

    For metric spaces, regarding set theory, you really just need to know what a map is. Some people also seem to have trouble digesting the concept of a set whose elements are sets i.e. what is usually called a family of a set.
  8. Jul 30, 2009 #7
    hmmm, I'm taking real analysis next semester and I know no set theory. Unless someone slipped it into my curriculum w/o telling me.

    I don't think you need it. My math department takes pretty good care of its undergrads, I think they'd make it a prereq or at least mention it if it was needed.
  9. Jul 30, 2009 #8
    Well, I don't think you need rigourous set theory and actualy I'm taking 424 next semester.
    I think if you've taken math 347 then you'll be just fine.
  10. Jul 30, 2009 #9
    Do you go to UIUC Jin? I took 348 with George Francis. He wasn't that great of a prof. There was no structure to the class and it seems like they just lumped the transfer students in there. We didn't even use D'Angelo West text.
  11. Jul 30, 2009 #10
    Hi jmnance. Thanks for posting the links to the actual classes you're taking, as this always helps in these discussions. I actually read them, and you should be okay. The real analysis mentions it will cover the basic topology for Rn (metric spaces), which is all you will need. Teaching metric spaces is usually one of the core topics in an introductory course to real analysis, so you don't need to already know this. All the set theory you need to know is very basic stuff like De Morgan's laws, but you will probably cover this in your class anyway.

    The 432 course is not even a pre-requisite for the 447 class. You mentioned that you know basic techniques of proof, and that is the main thing you will need.

    Do you know what books will be used? A good introductory to set theory, logic, functions, and basic analysis is Steven Lay, which may be of some help during the course, depending on the level.
    Last edited by a moderator: Apr 24, 2017
  12. Jul 30, 2009 #11
    Yeah and I'm gonna be a sophomore next semester.

    I don't know that prof,last semester I took Kevin Ford's 347 class.
    I've asked Math Department permission to take the honors section of real analysis,
    they told me if I'm getting straight A's in calculus sequence then I should be fine.
    So I assume the course is not that hard as you may expected.

    btw, are u a physics major?
  13. Jul 31, 2009 #12
    N!kofeyn- Thank you for that very constructive comment. I am a bit more at ease now :) we are using the book by Ross called introduction to real analysis.

    Jin8-I am a senior in the math department used to be a phys-math double major, but doing physics as a major burnt me out really quick and I prefer to just study it on my own, leisurely, as I did before I chose it as a major. I had completed all the engineering physics classes as well as classical mechanics and relativity I and II for physics majors. I talked to my advisor and he said that since I am going to grad school, I need to take the most difficult undergrad analysis class they offer- 447. I also asked about the honors real an. and he said that it was a step below 447. I'm not sure why 447 doesn't make a clear distinction between the two.
  14. Jul 31, 2009 #13
    No problem. As for the book, cool. I recently saw that book suggested on here and taking a look at it on Google Books it looks very good. I saw that you switched to math and are planning to go to graduate school. If this is your last year, and if you haven't already, I would try and take an abstract algebra and higher linear algebra course in addition to your real analysis and topology courses.
  15. Jul 31, 2009 #14
    my situation is unique really... I completed 4 semesters worth of credits at a community college prior to entering uiuc. I am the age of and have attended school the equivalent time of, a sophomore entering as a junior. However, credit-wise I am a junior entering as a senior. When I graduate, I will be a "super senior" (senior plus a semester), but actually, I will be graduating a semester early. I will have taken the following math classes prior to graduation:

    introduction to mathematical proofs
    applied linear algebra
    non Eulclidean geometry
    Probability theory (calculus based)
    intro to abstract algebra 1
    set theory and topology of metric spaces
    intro to abstract algebra 2
    Real Variables
    honors seminar
    compx variables
    graduate (500 level) abstract algebra
    published paper
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