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Set Theory Proof

  1. Feb 28, 2008 #1
    1. The problem statement, all variables and given/known data

    This is the problem stated verbatim. xo is supped to be x with a subscript o.

    Suppose that A is a set and there exists xo ε A for which lx-xol ≤ r. Is it necessarily true true that for all x,y εA, we will have lx-yl ≤2r?

    2. Relevant equations

    Well, this problem is just supposed to test our ability to formulate proofs, so I can't think of many helpful equations.

    I do know lxl < r is the same thing as -r<x<r

    3. The attempt at a solution

    I can't even think of how to start proving this. In desperation, I tried to think of a counterexample to prove this untrue, but none came to mind. It's also weird that r is such an arbitrary value or variable or something. Nothing is stated in the problem on what it's supposed to represent. I think it means any picked real number but I'm not sure.

    Any ideas?
  2. jcsd
  3. Feb 28, 2008 #2
    For clarity, x is supposed to be any element of A correct? If so, then think about using the triangle inequality if it's available.
  4. Feb 29, 2008 #3
    Ok sweet, I completely forgot about the triangle inequality (yes it is available.) However, Im not sure on how to implement it exactly.

    So lx+-xol≤ lxl +lxol by the triangle inequality.
    lx-yl≤ lxl+lyl ditto.

    SO now my equation has become this:

    if lxl+lxol≤r
    is it true that lxl + lyl≤2r.

    Because if I prove that, the desired proof follows?
    Last edited: Feb 29, 2008
  5. Feb 29, 2008 #4
    No, that's not true. Just because |x - xo| is less than or equal to r doesn't mean that |x| + |xo| is too. Really, think more intuitively about what |x - xo| being less than r says.
  6. Feb 29, 2008 #5
    I'm lost still, mainly beacuse of the confusion between x and xo and what r represents.

    Its the difference between a picked value and any other value in a set? sorry If I'm coming across as a total dimwit, but I'm not experiences any revelations here..

    Prove that the value of the difference between any two members of a set is always less than twice the value always greater that the difference between any member and one specified member of the same set?
  7. Feb 29, 2008 #6
    If |x-x0| < r you can think of it as saying that the disk of radius r about x0 encloses the set. What can you then conclude?
  8. Feb 29, 2008 #7
    R = RADIUS!

    I'm an idiot too often, sorry.

    So obviously this set can be represented by a circle, where each point is obvious less that a diameter's width away from every other point. diameter = 2r.

    I will say that mathdope is a misleading name because you've helped me a ton. Now I just need to ship that explanation up into a mathematical set logic proof rather than a verbal diagram proof, but I thinki should be good to go. Thanks a million
  9. Feb 29, 2008 #8

    A picture should help with the proof. Draw a circle with x and y at random points and x0 at the center. Then try to fit the triangle inequality in with your picture.
  10. Feb 29, 2008 #9
    Sorry to continue to be such a nag, but is there any way I could prove this without refering to the circle at all (my proffessor is crazy about not using diagrams). I understand that lx-yl represents the distance between any two points, which obviously can't exceed 2*r, but then I think I'm assuming the thing I want to prove. Its easy to explain the proof in the context of the circle idea, but without it I seem to stumble even more when I try to just tplainly use inequalities.

    Wow, this is the most hopeless I've even felt, and I'm on the dean's list of students too... I really hope I'm just really tired and not normally this stupid.
  11. Feb 29, 2008 #10
    Here's a tip that will get you through a good chunk of analysis problems: add 0 inside the absolute value sign in a "creative" way.
  12. Feb 29, 2008 #11
    Hopefully, this will be my last post.

    We have x,y as elements of A. Thus we know that


    and we know that

    based on the question.
    So we can add these together to get


    which is the same as

    lx-xol + lxo-yl≤2r

    By the triangle inequality, we know that

    l(x-xo)+(xo-y)l ≤ lx-xol + lxo-yl

    So we combine these to get

    l(x-xo)+(xo-y)l ≤ lx-xol + lxo-yl ≤ 2r

    Thus, l(x-xo)+(xo-y)l ≤ 2r
    Finally, lx-yl≤ 2r

    Please say thats doable, I want some sleep!
    Edit: I have done the bolded step before, even though it's awkward, but my teacher says it's fine in other proofs.
  13. Feb 29, 2008 #12
    Looks like you got the jist.
  14. Feb 29, 2008 #13
    you shouldn't apologize for not knowing something =)
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