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Homework Help: Show that the set S is Closed but not Compact

  1. Feb 5, 2015 #1
    1. The problem statement, all variables and given/known data

    Show that the set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2
    is closed but not compact.

    2. Relevant equations
    set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2

    3. The attempt at a solution
    I set x = 0 and then y = 0
    giving me
    [0,±√3] and [±√3,0] which means it is closed

    However, for it to be Compact, it needs to be Closed & Bounded.
    For it to be bounded, it must have both an upper and lower bound, which to me it appears to have?
    The bound for me are when x and y = ±√3

    Clearly I am wrong, given how the question is structured. Any ideas?
  2. jcsd
  3. Feb 5, 2015 #2
    You'll have to be more clear o. What exactly your set S is. ##{(x,y) \in R^2 : 2x^2+xy+y^2}## is what you've defined. But, I can plug any points x, y into that since you've not told us what the equation must satisfy.
    Last edited by a moderator: Feb 5, 2015
  4. Feb 5, 2015 #3
    Most correct, my omission.
    The function equals 3.
  5. Feb 5, 2015 #4
    OK, so if the function must equal 3, then x=0 and y=0 is not even in the set. Your definition of compact is closed and bounded. So you need to show that the set is closed but not bounded.
  6. Feb 5, 2015 #5

    Ray Vickson

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    Are you saying that your ##(x,y)## must satisfy ##2x^{2+xy+y^2} = 3##? That is the function you wrote. Or, did you mean ##2 x^2 + xy + y^2 = 3##?
    Last edited: Feb 5, 2015
  7. Feb 5, 2015 #6
    Sorry, I responded on my phone so I'm not sure how the LaTex parts turned out.
  8. Feb 6, 2015 #7
    I need to apologise to everyone . . . immediately after I posted, I needed to run to lectures . . . I never noticed the error in my post, and wasn't able to correct it from my mobile phone. Really did not mean to waste your time. Stated correctly, it should read as:

    Show that the set S of all (x,y) ∈ ℝ2 such that x2+xy+y2 = 3
    is closed but not compact.

    (Have been trying to correct the original post, but doesn't seem possible?)
  9. Feb 6, 2015 #8
    Please see my apology below.
    You are correct, it should read:

    Show that the set S of all (x,y) ∈ ℝ2 such that x2+xy+y2 = 3
    is closed but not compact.
  10. Feb 6, 2015 #9
    Yes, which I think I did.
    I tried to show it was closed, but don't know how to show that it is bounded or not, as the case may be?
  11. Feb 6, 2015 #10
  12. Feb 6, 2015 #11
    ##x=y=\sqrt(3)## Is not in the set, nor is 0. You'll need to show that they are upper and lower bounds.

    Another definition of bounded is that you can find a ball in which the whole set is contained in.
  13. Feb 6, 2015 #12


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  14. Feb 8, 2015 #13


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    Let u= x+ y and v= x- y. Then x= (u+ v)/2 and y= (u- v)/2. Replace x and y in the equation. That will eliminate the "xy" term and it is easy to see what kind of conic you have.
  15. Feb 9, 2015 #14


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    I think there's a further error in the problem statement: S as you have defined it is actually compact.
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