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Math Help - Defining Big Intersection

  1. #1
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    Defining Big Intersection

    Hi guys.
    I completely understand the definition for the intersection between two or more sets, but this definition just does not make sense.



    If M is a set of sets. For all A which are an element of M, x is an element of A. Isn't that the exact same thing as the union? Am I reading this wrong? I don't see how this definition covers the fact that x HAS to appear in EVERY A. It just states that x is an element of A, not every A.
    Last edited by ShadowKnight8702; December 14th 2013 at 01:58 PM.
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  2. #2
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    Re: Defining Big Intersection

    I hope I dont say anything wrong, but this is my take on it:

    You have a non empty set M. The intersection of M is the set of elements, which are in EVERY set of elements of M. Thats why there is a ∀. You just look specifially at the intersection only and not any other parts of sets. I guess thats what confuses you.
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  3. #3
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    Re: Defining Big Intersection

    Thanks. I was just reading the statement wrong, separating the two properties.
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  4. #4
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    Re: Defining Big Intersection

    Quote Originally Posted by ShadowKnight8702 View Post

    If M is a set of sets. For all A which are an element of M, x is an element of A. Isn't that the exact same thing as the union? Am I reading this wrong? I don't see how this definition covers the fact that x HAS to appear in EVERY A. It just states that x is an element of A, not every A.
    It is far better and preferred to say "if M is a collection of sets".
    \bigcap M  = \left\{ {x : (\forall A \in M)\left[ {x \in A} \right]} \right\} and \bigcup {M = \left\{ x: (\exists A \in M)\left[ {x \in A} \right]} \right\}

    Note in the first it is \forall A \in M but in the second it is \exists A \in M.
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