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Math Help - Union of untersections

  1. #1
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    Lightbulb Union of untersections

    Hi. I was wondering if you could help me solve this. I've tried drawing it but it doesn't work, and anyway, I want to be able to solve it in a more elegant manner.

    \bigcap_{m=1}^{+ \infty}  \bigcup_{n=1}^{+ \infty} (-\frac {m}{n}, \frac {n}{m})

    Thank you.
    Last edited by wilhelm; January 7th 2013 at 09:16 PM.
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  2. #2
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    Re: Union of untersections

    Hey wilhelm.

    I would consider the smallest end-points in magnitude from the origin for each value of m.

    The smallest in the negative is -1 and the smallest on the positive side is unbounded (since n/m -> infinity).

    So with this you are going to have (-1,infinity) intuitively with the above reasoning (which may be wrong and if it is please point it out).
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  3. #3
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    Re: Union of untersections

    Quote Originally Posted by wilhelm View Post
    Hi. I was wondering if you could help me solve this. I've tried drawing it but it doesn't work, and anyway, I want to be able to solve it in a more elegant manner.
    \bigcap_{m=1}^{+ \infty}  \bigcup_{n=1}^{+ \infty} (-\frac {m}{n}, \frac {n}{m})
    \bigcap_{m=1}^{+ \infty}\left[ \bigcup_{n=1}^{+ \infty} (-\frac {m}{n}, \frac {n}{m})\right]

    Note that I added some [] to emphasize the union is done first. With union we "never loose any elements"

    Thus for each M the union yields (-M,\infty).

    Recall that with intersection we get only the common part.

    So what is the common part of all of those unions?
    Thanks from wilhelm
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  4. #4
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    Re: Union of untersections

    Is the answer (-1, + \infty) ?
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