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Subdivisions/Refinement Proof

  1. Feb 11, 2013 #1
    I understand this isn't a homework area but there is always so much more traffic in this forum rather than the homework. All I'm looking for is a clarification that my ideas to prove both parts are in fact accurate.


    1. The problem statement, all variables and given/known data

    If each of D1 and D2 is a subdivision of [a,b], then...
    1. D1 u D2 is a subdivision of [a,b], and
    2. D1 u D2 is a refinement of D1.


    2. Relevant equations

    **Definition 1: The statement that D is a subdivision of the interval [a,b] means...
    1. D is a finite subset of [a,b], and
    2. each of a and b belongs to D.


    **Definition 2: The statement that K is a refinement of the subdivision D means...
    1. K is a subdivision of [a,b], and
    2. D is a subset of K.



    3. The attempt at a solution

    My problem is that I've taken a lot of logic courses in the past so when I see the union of two variables I only need to prove that one is actually true. In this particular situation both are true so its obvious but I don't know how to state that fact.

    For the 2nd part of the proof, wouldn't I just say that D1 is a subset of itself, and its already given that D1 is a subdivision of [a,b]? It just seems too easy...


    I also had questions about proofs I've already turned in that I did poorly on but I didn't want to flood this place with questions.
     
  2. jcsd
  3. Feb 11, 2013 #2

    mfb

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    Staff: Mentor

    This is not a logical "or".

    Consider a simple setup:
    A={1}
    B={2}
    The statement "the set has exactly one element" is true for both A and B, but not for their union C={1,2}.

    To show that E := D1 u D2 is a subdivision, you have to show that E is a finite subset of [a,b] and a and b are in E.

    If you prove "1." first, that works.
     
  4. Feb 11, 2013 #3
    See I was thinking more along the lines if we had x an element of A then its simple enough to just say X is in A u B?

    Because it doesn't matter if X is in B since we can just add another set, regardless its still in A. I'm just lost how to incorporate the union into the proof, I guess I'll spend more time on that.

    Haha. See within 5 minutes I get a reply in this one and not the homework =P. Thank you!
     
  5. Feb 11, 2013 #4

    mfb

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    ##x \in A \Rightarrow x \in (A \cup B)##, and I think you can use this as it is very elementary.
     
  6. Feb 11, 2013 #5
    I didn't see the drop down menu for the element, union, and implication arrows. Where are those?
     
  7. Feb 12, 2013 #6

    mfb

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    Staff: Mentor

    They are written with the [itex]-tags and LaTeX codes. See the FAQ entry for details, or quote my post to see its code.
     
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