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No. of elements

  1. May 7, 2015 #1
    1. The problem statement, all variables and given/known data
    If I am given ##n(A)## and ##n(B)## for two sets A and B, and also provided with ##n(A\cap B)^2##. We are supposed to find ##n((AXB) \cap (BXA))##.
    2. Relevant equations
    My teacher said that the formula for ##n((AXB) \cap (BXA)) = n(A \cap B)^2##. I am not sure how do you get to this result.

    3. The attempt at a solution
    So my question is, how do you prove (or derive the relation)-
    ##n((AXB) \cap (BXA)) = n(A \cap B)^2##
     
    Last edited by a moderator: May 8, 2015
  2. jcsd
  3. May 7, 2015 #2

    SammyS

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    Clarification

    By ##\displaystyle\ AXB\ ## do you mean the direct product ##\displaystyle\ A\times B\ ## ?

    Also is ##\displaystyle\ n(A\cap B)^2\ ## actually the square of the number ##\displaystyle\ n(A\cap B)\ ## ?

    .
     
    Last edited by a moderator: May 8, 2015
  4. May 7, 2015 #3
    Yes. Sorry, I don't know how to get the thinner cross. And yes, for the second case too.
     
  5. May 7, 2015 #4

    SammyS

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    \times

    ## \text{\times gives you}\ \times ##
     
  6. May 8, 2015 #5
    You must show that ## (A\times B) \cap (B\times A) = (A\cap B)\times (A\cap B) ##. The answer is then straightforward if you know how to calculate the cardinal of a cartesian product.
    More generally, ## (A\times B) \cap (C\times D) = (A\cap C) \times (B\cap D) ## (There is no such rule for a union)



    btw: the question makes sense only if A and B are finite sets
     
    Last edited: May 8, 2015
  7. May 8, 2015 #6
    Thanks!
     
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