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Thread: 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.
    $\displaystyle (A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))$

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

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

    And the property,
    $\displaystyle X\cap Y\subseteq X$

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

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

    Thus, (by transitive propetry),
    $\displaystyle
    (A\cap B\cap C)\subseteq ((B\cap (A\cup C))\cup (C \cap A))
    $
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