Results 1 to 4 of 4

Math Help - set theory operation

  1. #1
    Member
    Joined
    Aug 2007
    Posts
    239

    Smile set theory operation

    We want to prove that for each  A,B \in \mathcal{P}(X) , there exists a unique set  C = A \Delta B such that  A \Delta C = B .

    So  A \Delta C = B \Rightarrow (A-C) \cup (C-A) = B \Rightarrow A-C \subseteq B and  C-A \subseteq B \Rightarrow  C = (A-B) \cup (B-A) \Rightarrow C = A \Delta B ?

    Obviously  C = A \Delta B because  A \Delta (A \Delta B) = (A \Delta A) \Delta B = \emptyset \Delta B = B (by associative law). But that only proves existence and not uniqueness.

    Thanks
    Last edited by shilz222; August 18th 2007 at 09:14 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1
    You should know that the symmetric difference operation is associative:
    \left( {K\Delta L} \right)\Delta M = K\Delta \left( {L\Delta M} \right).

    If C = A\Delta B then consider this:
    A\Delta C = A\Delta \left( {A\Delta B} \right) = \left( {A\Delta A} \right)\Delta B = \emptyset \Delta B = B. We have shown the property.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2007
    Posts
    239
    Yes thats what I did in my first post.

    But does that suffice to show that  C = A \Delta B ?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1
    Uniqueness
    If A\Delta L = B then \left( {A\Delta L} \right)\Delta B = \emptyset.
    But \left( {A\Delta L} \right)\Delta B = \left( {A\Delta B} \right)\Delta L = \emptyset \quad  \Rightarrow \quad L = A\Delta B
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. row operation
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 8th 2011, 02:19 PM
  2. Operation @
    Posted in the Advanced Math Topics Forum
    Replies: 5
    Last Post: April 2nd 2010, 12:10 AM
  3. Operation
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 24th 2010, 08:20 PM
  4. binary operation
    Posted in the Algebra Forum
    Replies: 3
    Last Post: December 3rd 2008, 10:27 AM
  5. row operation
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 17th 2007, 03:39 AM

Search Tags


/mathhelpforum @mathhelpforum