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Math Help - Proving a Set

  1. #1
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    Proving a Set

    Note: ^ = "intersects"; U = "union"

    Prove that A ^ B ^ C "is a subset of" D, where D = (B ^ (A U C)) U (C ^ A)

    Obviously, when drawing a Venn diagram, they all will meet at the very middle. How can I prove this is true, in a formal proof.
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  2. #2
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    Quote Originally Posted by PhilipJ
    Note: ^ = "intersects"; U = "union"

    Prove that A ^ B ^ C "is a subset of" D, where D = (B ^ (A U C)) U (C ^ A)

    Obviously, when drawing a Venn diagram, they all will meet at the very middle. How can I prove this is true, in a formal proof.
    (A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))

    If A\cap B\cap C=\{\} there is nothing to prove because the empty set is subset of every set.

    If not then, \exists x such as x\in A,B,C
    Notice that x\in ((B\cap (A\cup C)) because x\in B and x\in A\cup C.
    Thus,
    x\in ((B\cap (A\cup C))\cup (C \cap A))
    Thus, any element of (A\cap B\cap C) is element of ((B\cap (A\cup C))\cup (C \cap A)). Thus, by definition,
    (A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))
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  3. #3
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    Also, using the "transitive" property of subsets:
    X\subseteq Y and Y \subseteq Z then, X\subseteq Z

    And the property,
    X\cap Y\subseteq X

    And finally the property,
    X\subseteq X\cup Y
    ---------------
    Thus,
    B\cap (A\cap C)\subseteq A\cap C
    Because X=B,Y=A\cap C

    And,
    A\cap C\subseteq ((B\cap (A\cup C))\cup (C \cap A))
    Because X=(C \cap A)),Y=((B\cap (A\cup C))

    Thus, (by transitive propetry),
    <br />
(A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))<br />
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