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Math Help - Proof

  1. #1
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    Proof

    Show (A U B) X C = (A X C) U (A X B)

    And Show that AXC and AXB are disjoint

    Any advice?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by jzellt View Post
    Any advice?
    why, yes

    Show (A U B) X C = (A X C) U (A X B)
    here's a hint. show that (A \cup B) \times C \subseteq (A \times C) \cup (A \times B) and then show (A \times C) \cup (A \times B) \subseteq (A \cup B) \times C. this will prove the equality

    And Show that AXC and AXB are disjoint
    there is no way we can know this without knowing how A, B, and C relate to each other. clearly if B = C or either is a subset of the other, then this is not the case.

    and please start using more descriptive titles than just "proof"
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  3. #3
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    Quote Originally Posted by jzellt View Post
    Show (A U B) X C = (A X C) U (A X B)

    And Show that AXC and AXB are disjoint

    Any advice?
    The above identity is wrong as the following counter example shows:

    Let A={1,2},B={3,4} ,C={a}. AND:

    AUB= {1,2}U{3,4} = {1,2,3,4}

    (AUB)xC = {1,2,3,4}x{a} = {(1,a),(2,a),(3,a),(4,a)}

    But:

    AxC= {1,2}x{a} = {(1,a),(2,a)}

    Also AxB = {1,2}x{3,4} = {(1,3),(1,4),(2,3),(2,4)},and

    (AxC)U(AxB)= {(1,a),(2,a)}U {(1,3),(1,4),(2,3),(2,4)}={(1,a),(2,a),(1,3),(1,4) ,(2,3),(2,4)}

    Hence:  (AUB)xC\neq(AxC)U(AxB)
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  4. #4
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    Quote Originally Posted by jzellt View Post
    Show (A U B) X C = (A X C) U (A X B)

    And Show that AXC and AXB are disjoint

    Any advice?
    I strongly suspect this was supposed to be (A U B) X C = (A X C) U (B X C).
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