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Math Help - union of collections

  1. #1
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    union of collections

    I have a simple question, if we have two collection of sets, instead of sets, A and B, how is the intersection of A and B defined, is it defined as
    {a intersection b:a a set in A and b a set in B}

    thx
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  2. #2
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    Re: union of collections

    Quote Originally Posted by sung View Post
    I have a simple question, if we have two collection of sets, instead of sets, A and B, how is the intersection of A and B defined, is it defined as {a intersection b:a a set in A and b a set in B}
    From that description it is very hard to understand your question.
    Say you have two collections \mathcal{A}=\{P,Q,R\}~\&~\mathcal{B}=\{U,V,X,Y\} of sets.
    Show what you are saying about \mathcal{A}\cap\mathcal{B}.
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  3. #3
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    Re: union of collections

    A \cap B={ P \cap U, P \cap V,...}

    So can we define this as
    A \cap B={ I_{1} \cap I_{2};for I_{1} \in A and I_{2} \in B}
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  4. #4
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    Re: union of collections

    Quote Originally Posted by sung View Post
    A \cap B={ P \cap U, P \cap V,...}
    So can we define this as
    A \cap B={ I_{1} \cap I_{2};for I_{1} \in A and I_{2} \in B}
    So you want a collection of set \{G\cap H:G\in\mathcal{A}\text{ and }H\in\mathcal{B}\}
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  5. #5
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    Re: union of collections

    yes is that the correct definition for the intersection of two collections?
    So how is the intersection of two collections defined?
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  6. #6
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    Re: union of collections

    Quote Originally Posted by Plato View Post
    So you want a collection of set \{G\cap H:G\in\mathcal{A}\text{ and }H\in\mathcal{B}\}
    Quote Originally Posted by sung View Post
    So how is the intersection of two collections defined?
    I have no idea what you mean by that question.
    This is a collection of sets \{G\cap H:G\in\mathcal{A}\text{ and }H\in\mathcal{B}\}

    J\in\mathcal{A}\cap\mathcal{B}\iff J\in\mathcal{A}\text{ and }J\in\mathcal{B}.

    \bigcup\limits_{G \in A\;\& \;H \in B} {\left\{ {G \cap H} \right\}} is the set of elements which are in the intersection of a set in \mathcal{A} and some set in \mathcal{B}.
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  7. #7
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    Re: union of collections

    The definition of the intersection of two sets is the same regardless of the nature of the elements of those two sets: whether they are numbers, other sets, etc.
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  8. #8
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    Re: union of collections

    Quote Originally Posted by emakarov View Post
    The definition of the intersection of two sets is the same regardless of the nature of the elements of those two sets: whether they are numbers, other sets, etc.
    While I agree that is the case and said so when I posted
    Quote Originally Posted by Plato View Post
    J\in\mathcal{A}\cap\mathcal{B}\iff J\in\mathcal{A}\text{ and }J\in\mathcal{B}.
    I think that the OP had something more in mind.
    Such as a generalize intersection.
    That is why I asked for clarification.
    But I did not really get what I thought I would.
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