Results 1 to 6 of 6

Math Help - [SOLVED] set proof help

  1. #1
    Junior Member
    Joined
    Sep 2007
    Posts
    30

    [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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570

    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
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jul 2009
    Posts
    24
    hello running-gag
    ..If then ..
    how can you say like that..
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    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
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jul 2009
    Posts
    24
    ok.thank you
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] direct proof and proof by contradiction
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: February 27th 2010, 10:07 PM
  2. [SOLVED] Please help with proof
    Posted in the Calculus Forum
    Replies: 6
    Last Post: February 2nd 2010, 11:11 AM
  3. [SOLVED] How Much of a Proof is this
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: July 12th 2009, 11:21 AM
  4. [SOLVED] Proof GCD and LCM
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: March 7th 2009, 09:45 AM
  5. [SOLVED] proof
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 10th 2009, 05:04 PM

Search Tags


/mathhelpforum @mathhelpforum