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Math Help - Field Theory Proof -- Please Help

  1. #1
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    Field Theory Proof -- Please Help

    Let X,Y,Z be sets. Prove that X union (Y intersecting Z) = (X union Y) intersecting (X union Z).

    The proof would start off by showing that each side is a subset of the other, I believe... But there would be several cases? I'm not really sure where to go with this.. Any help would be wonderful.
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  2. #2
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    X\cup(Y \cap Z)=(X\cup Y)\cap(X \cup Z)

    That's the deMorgan law proof for sets or whatever.

    You take the left side or the right side, write out what it means to be a union and stuff just a base definition.

    So say the left side:

    Y \cap Z) = \left \{ x: x \in Y and \text{  } x \in Z \right \}

    Let's let E = Y \cap Z

    X \cup E = \left \{ x: x \in X or \text{  } x \in E \right \}

    So combine it: (I don't really remember what I did when I did this the first time but it gets kinda messy)

    =\left \{ x: x \in X or \text{  } (x \in Y and \text{  } x \in Z )\right \}

    Then you do the same to the right side and you can see they're equal.
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