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Math Help - Simple Set Theory Theorem

  1. #1
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    Simple Set Theory Theorem

    Hello,

    I am trying to prove that

    A is a subset of B
    if and only if
    the intersection of A and B is the set A.

    It it seems obvious from the definitons but I am having trouble formalizing the proof.

    In the forward direction, we know an element x in A implies x in B, and that the intersection of A and B is just all the elements y such that y is in A AND y is in B...

    So it seems obvious to me that A intersection B is just A, but how can I state it formally? Similarily for the backwards direction.

    Thanks!
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  2. #2
    MHF Contributor

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    I do it one way. Suppose that A\cap B=A.
    If x\in A then it follows that x\in A\cap B implying that x\in B.
    So A\subseteq B.
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