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Homework Help: Element proof for sets A,B,C

  1. Jul 16, 2014 #1
    1. The problem statement, all variables and given/known data


    [tex]A \times (B \cap C) = (A \times B) \cap (A \times C) [/tex]

    3. The attempt at a solution

    Let [tex]x \in A[/tex] and [tex]y \in B \cap C \rightarrow y \in B \wedge y \in C[/tex]

    now [tex] \exists (x,y) \in A \times (B \cap C) [/tex]

    so [tex](x,y) \in A \times B \wedge (x,y) \in A \times C[/tex]

    thus [tex](x,y) \in (A \times B) \cap (A \times C) [/tex]

    therefore [tex]A \times (B \cap C) = (A \times B) \cap (A \times C) [/tex]
  2. jcsd
  3. Jul 17, 2014 #2


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    A,B,C could all be the empty set, in which case your third line is incorrect, there may not exist such (x,y).

    I would use set-builder notion to show that they are the same.
  4. Jul 17, 2014 #3
    If they are all the empty set then thats pretty trivial and not interesting. And all the same? As in A =B=C?
  5. Jul 17, 2014 #4


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    I mean the left and right-hand sides, you want to show that they are the same set.
  6. Jul 17, 2014 #5


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    I addition to what verty pointed out:

    You have only done half of the proof.

    You showed that the left hand side is a subset of the right hand side.
  7. Jul 17, 2014 #6

    Ah yes. I went back and showed it goes both ways. Thanks
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