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Cartesian Product Proof

  1. Apr 17, 2007 #1
    Hi all...

    1. The problem statement, all variables and given/known data

    Let A, B be non-empty sets, proof that A x B = B x A iff A = B


    2. Relevant equations

    A x B = Cartesian Product
    iff = if and only if
    ^ = and


    3. The attempt at a solution

    Let (x,y) є A x B = B x A
    iff (x,y) є (A X B) ^ (x,y) є (B x A)
    iff (x є A ^ y є B) ^ (x є B ^ y є A)
    iff (x є A ^ y є A) ^ (x є B ^ y є B)
    iff (x,y) є A ^ (x,y) є B
    iff (x,y) є A = B

    Its that right?

    Also, if one of the sets if empty, will the statement hold?

    Thanks guys!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 17, 2007 #2

    Dick

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    Your proof is correct in essence, but hugely confusing and ungrammatical. (x,y) for x in A and y in A is not an element of A. It's an element of AxA. First prove if A=B then AxB=BxA. That's pretty easy, right? Now prove if AxB=BxA then A=B. It's actually easiest (and much more clear) to prove this by contradiction. And if one of the sets is empty then the cartesian product is empty. Does that make that case easy?
     
  4. Mar 23, 2009 #3
    Hi, I do not understand the proof and I see that my proof is inadequate. How would you do this by contradiction? And does AxA need to be in the proof? This is what I did:

    assume AxB=BxA
    let x be an element of A,B ^ y be an element of A,B
    (x,y) is an element of A ^ (x,y) is an element of B
    so A=B
     
  5. Mar 23, 2009 #4

    Dick

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    To prove two sets are equal, you want to prove every element of one is an element of the other. Start with your assumption AxB=BxA. Pick any x in A and any y in B. Then (x,y) is an element of AxB. But since AxB=BxA that mean (x,y) is also an element of BxA. Hence?
     
  6. Mar 23, 2009 #5
    Thanks for responding and helping me, but I'm not sure if I'm following, here is what I get: I should show

    assume AxB=BxA
    let x be an element of A ^ y be an element of B
    (x,y) is an element of AxB
    if yes then (x,y) is an element of BxA
    so AxB=BxA
    so A=B
     
  7. Mar 23, 2009 #6

    Dick

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    No, no. You assumed AxB=BxA. You don't conclude it. If (x,y) is an element of BxA then x is an element of B and y is an element of A. But remember x was ANY element of A and y was ANY element of B. So A=B BECAUSE any element of A is an element of B and vice-versa.
     
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