Results 1 to 4 of 4

Math Help - Sets!

  1. #1
    Member
    Joined
    Aug 2007
    Posts
    239

    Sets!

    Prove that  A^{c} \subseteq B \Leftrightarrow A \cup B = U (universal set). From a previous part of the problem I already proved that  B \subseteq A^{c} \Leftrightarrow A \cap B = \emptyset . The problem then says to take complements to deduce the former result.

    So I said that  A^{c} = B and so  A \cup B = U by definition. In the other direction I am given that  A \cup B = U . I tried using DeMorgans Laws:  (A \cup B)^{c} = A^{c} \cap B^{c} = \emptyset and so  A^{c} \subseteq B ? Is this correct? Did I do what the hint told me?


    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1
    Given that A^c  \subseteq B then we know that
    U = A \cup A^c  \subseteq A \cup B \subseteq U or A \cup B = U.

    Given that A \cup B = U then
    A^c  \cap \left( {A \cup B} \right) = A^c \quad  \Rightarrow \quad A^c  \cap B = A^c \quad  \Rightarrow \quad A^c  \subseteq B.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2007
    Posts
    239
    What about my use of the De Morgan LAws? IS that incorrect?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1
    Quote Originally Posted by shilz222 View Post
    What about my use of the De Morgan LAws? IS that incorrect?
    No, what you did is correct.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Open sets and sets of interior points
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 9th 2011, 03:10 AM
  2. Metric spaces, open sets, and closed sets
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 16th 2011, 05:17 PM
  3. Replies: 9
    Last Post: November 6th 2010, 12:47 PM
  4. Approximation of borel sets from the top with closed sets.
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: February 18th 2010, 08:51 AM
  5. how to show these sets are 95% confidence sets
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: February 11th 2009, 09:08 PM

Search Tags


/mathhelpforum @mathhelpforum