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Math Help - [SOLVED] set proof help

  1. #1
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    [SOLVED] set proof help

    Can anyone prove or disprove
    X \cap Z \subseteq Y \Leftrightarrow (X - Y) \cup (Y-Z) \subseteq Z^c

    I think you have to split it up into two parts, but whenever I do it one way I can't get the same exact argument to go back the other way.
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  2. #2
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    Quote Originally Posted by horan View Post
    Can anyone prove or disprove
    X \cap Z \subseteq Y \Leftrightarrow (X - Y) \cup (Y-Z) \subseteq Z^c

    I think you have to split it up into two parts, but whenever I do it one way I can't get the same exact argument to go back the other way.
    Hi

    Suppose that X \cap Z \subseteq Y
    Let's prove that (X - Y) \cup (Y-Z) \subseteq Z^c

    Let x \in (X - Y) \cup (Y-Z)
    We have to prove that x \in Z^c

    x \in (X - Y) \cup (Y-Z) means that or x \in (Y-Z) or x \in (X - Y)
    If x \in (Y-Z) then x \notin Z and therefore x \in Z^c => OK
    If x \in (X - Y) then x \in X and x \notin Y. If x \in Z then x \in X \cap Z. But X \cap Z \subseteq Y therefore x \in Y which is not possible since x \notin Y. Threfeore x \notin Z.
    In both cases x \in Z^c.
    Therefore (X - Y) \cup (Y-Z) \subseteq Z^c.
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  3. #3
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    Set Theory Proof

    Hello horan
    Quote Originally Posted by horan View Post
    Can anyone prove or disprove
    X \cap Z \subseteq Y \Leftrightarrow (X - Y) \cup (Y-Z) \subseteq Z^c

    I think you have to split it up into two parts, but whenever I do it one way I can't get the same exact argument to go back the other way.
    running-gag has proved that X \cap Z \subseteq Y \Rightarrow (X - Y) \cup (Y-Z) \subseteq Z^c. Here's the proof of X \cap Z \subseteq Y \Leftarrow (X - Y) \cup (Y-Z) \subseteq Z^c.

    Suppose that (X - Y) \cup (Y-Z) \subseteq Z^cand that x \in X\cap Z. Therefore we must show that x \in Y.

    x \in X \cap Z \Rightarrow x \in Z

    \Rightarrow x \notin Z^c

    \Rightarrow x \notin (X - Y) \cup (Y-Z)

    \Rightarrow x \notin (X - Y)

    But x \in X

    Therefore x \in Y

    Grandad
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  4. #4
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    hello running-gag
    ..If then ..
    how can you say like that..
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  5. #5
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    Hello doresa
    Quote Originally Posted by doresa View Post
    hello running-gag

    ..If then ..

    how can you say like that..
    You're just quoting part of the whole proof here. Obviously you can't say that, in general, x \in Z \Rightarrow x \in X \cap Z.

    But the bit you're quoting is from a line that starts: If x \in (X-Y)... In other words, it pre-supposes that x \in X. Therefore if in addition x \in Z, then x \in X \cap Z.

    Is that OK now?

    Grandad
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  6. #6
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    ok.thank you
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