Results 1 to 4 of 4

Math Help - How to prove using cartesian product

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    13

    How to prove using cartesian product

    How can one prove that for all sets A and B, if A and B are nonempty and A X B = B x A, then A = B.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by brudman View Post
    How can one prove that for all sets A and B, if A and B are nonempty and A X B = B x A, then A = B.
    Start with (x,y)\in A\times B implies that x\in A~\&~y\in B.
    If A\times B=B\times A what can be said of x~\&~y?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2009
    Posts
    13
    x = y ??
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,513
    Thanks
    769
    x = y ??
    No, x and y are even from different sets (like apples and oranges).

    OK, so you need to prove that A = B. It is sufficient to show that every element of A is an element of B, and conversely. Let's prove the first part.

    What is the standard beginning of a proof of a statement of the following form: "Every such-and-such object has such-and-such property"? It's "Let's fix any such-and-such object". Write in symbols what this means here.

    Next, B is nonempty, so choose some element. Now we have one element from A, one from B. What can we form from them?

    OK, now A x B = B x A, so this object we formed is also in B x A. Now use Plato's remark. This should conclude the proof that A is a subset of B. The opposite inclusion is similar.
    Last edited by emakarov; November 15th 2009 at 01:48 PM. Reason: Spelling
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: January 9th 2013, 06:14 AM
  2. Cartesian product of A*A*A*A
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 14th 2011, 06:29 AM
  3. Replies: 5
    Last Post: April 12th 2010, 02:16 PM
  4. Cartesian product
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: August 27th 2009, 08:51 AM
  5. Cartesian Product
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: June 3rd 2009, 10:33 AM

Search Tags


/mathhelpforum @mathhelpforum