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Math Help - Finding the (countable) union of these events

  1. #1
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    Finding the (countable) union of these events

    I'm stumped on a few problems here. They are both very similar, and so I will ask just one (should be able to get the other one if I can get this one)

    Suppose that for i= 1,2,3,... the set Ai is the set of real numbers from (2/i) to (3-(1/i)) inclusive. What is the countable union of these events?

    What i've attempted so far:

    A1 = [2,2] = { 2 }
    A2 = [1, 5/2] = { x | XeR, 1<=X<=5/2}
    .
    ..
    ...
    Ai = [(2/i), (3-(1/i))] inslusive

    So in order to find the countable union of these events, i've used the fact that as i increases, the range the set covers increases, and since each set/event contains the previous one, the union of the set will grow towards A = (0,3) (exclusive) as i tends to infinity.

    I'm not sure if this is correct reasoning, or if there is a better method for attempting these problems. Also, the set A = (0,3) for real numbers is uncountably infinite.. isn't this a contradiction somewhere?
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  2. #2
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    Re: Finding the (countable) union of these events

    Quote Originally Posted by mkelly09 View Post
    So in order to find the countable union of these events, i've used the fact that as i increases, the range the set covers increases, and since each set/event contains the previous one, the union of the set will grow towards A = (0,3) (exclusive) as i tends to infinity.
    This is correct. It's always possible to provide more details, but I am not sure it is necessary.

    Quote Originally Posted by mkelly09 View Post
    Also, the set A = (0,3) for real numbers is uncountably infinite.. isn't this a contradiction somewhere?
    Each A_i for i > 1 is a nonempty interval, i.e., is uncountably infinite, so it's not surprising that the union is uncountably infinite.
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