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Math Help - Proving union and intersection of indexed family of sets

  1. #1
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    Proving union and intersection of indexed family of sets

    I have to prove that the union for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-infinity,1).
    And that the intersection for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-1,0].

    I've asked the professor twice now for help and he hasn't been able to explain it at all.
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    Re: Proving union and intersection of indexed family of sets

    show inclusions in both directions.
    for the intersection :what x is between -n and 1/n for all n?
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    Re: Proving union and intersection of indexed family of sets

    Quote Originally Posted by skippenmydesign View Post
    I have to prove that the union for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-infinity,1).
    And that the intersection for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-1,0].
    I think you are confused by the notation.
    Let D_n  = \left( { - n,\frac{1}{n}} \right).
    Now consider two examples: D_3  \cup D_1  = \left( { - 3,1} \right)\;\& \;D_3  \cap D_1  = \left( { - 1,\frac{1}{3}} \right)

    Now let's do an indexed example:
    \bigcup\limits_{k = 1}^{100} {D_k }  = \left( { - 100,1} \right)\;\& \;\bigcap\limits_{k = 1}^{100} {D_k }  = \left( { - 1,\frac{1}{{100}}} \right)

    Note how the right and left endpoints are limits.
    Do you see how it works?
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    Re: Proving union and intersection of indexed family of sets

    Ok, I'm not familiar with latex so I can't get this into symbolic form but I'm going to attach a picture of what the question is. I have to prove it using the definition that for two things to be equal they each have to be a subset of the other.
    Attached Thumbnails Attached Thumbnails Proving union and intersection of indexed family of sets-1104022055.jpg  
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    Re: Proving union and intersection of indexed family of sets

    Quote Originally Posted by skippenmydesign View Post
    Ok, I'm not familiar with latex so I can't get this into symbolic form but I'm going to attach a picture of what the question is. I have to prove it using the definition that for two things to be equal they each have to be a subset of the other.
    Frankly I have no idea why you posted exactly what you posted exactly what I had already posted.
    \lim _{n \to \infty } \bigcup\limits_{k = 1}^n {D_k }  = \left( { - \infty ,1} \right)\;\& \,\;\;\lim _{n \to \infty } \bigcap\limits_{k = 1}^n {D_k }  = \left( { - 1,0} \right]

    BTW: Why not learn to use LaTeX?
    Last edited by Plato; November 4th 2012 at 05:15 PM.
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    Re: Proving union and intersection of indexed family of sets

    because I need a proof for the answer? not the answer itself? I know the answer but not the proof for it.
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    Re: Proving union and intersection of indexed family of sets

    Also like I said I know from talking to the professor that the proof needs to use the definition of for things to be equal they have to be subsets of each other. I hope that is clear.

    Also I might learn latex sometime but not tonight!
    Last edited by skippenmydesign; November 4th 2012 at 05:37 PM.
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  8. #8
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    Re: Proving union and intersection of indexed family of sets

    try proving this "in-between" step:

    for the first problem-

    show that if k < m, that (Dk)U(Dm) = (-m,1/k). what is the smallest k can be, and what is the largest m can be?

    for the second problem-

    show that for k < m that (Dk)∩(Dm) = (-k,1/m). again: how small can k be, and how large can m be?

    (the answer for k in both cases should be "easy". answering for m might take a little thought).

    try drawing a picture with k = 3, and m = 4. draw another one with k = 1, and m = 10. what do you notice?
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