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Math Help - [SOLVED] Beginner Type Proof: subsets

  1. #1
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    [SOLVED] Beginner Type Proof: subsets

    Let A, B, C be sets..

    Prove that (AUB)∩ C is contained in AU(B∩ C)

    I know how to show this using the venn diagram, and one professor I had told us just drawing the venn diagram is proof enough, but I think this professor wants a more formal proof.

    My idea is to maybe let x belong to (AUB)∩ C and then this is like (A∩ C)U(B∩ C) and then A∩ C contains AU(B∩ C), but I feel like I am just running in circles.

    Thanks for any help.
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  2. #2
    Super Member Aryth's Avatar
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    Assume that:

    x \in (A \cup B)\cap C

    This means that:

    x \in [(A \cap C)\cup (B \cap C)]

    If we call (A \cap C) m and (B \cap C) n then we can rearrange them to get:

    x \in (n \cup m) = [(B \cap C) \cup (A \cap C)]

    Now, we expand yet again:

    x \in [((B \cap C)\cup A) \cap ((B \cap C) \cup C)]

    If x is in the intersection of the two, then they both have x in common, which means that:

    [x \in (B \cap C)\cup A] \cap [x \in (B \cap C)\cup C]

    This is sufficient to prove that:

    x \in (B \cap C)\cup A
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