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Help With Find The Cardinality of a Power Set of a Cartesian Product

  1. Mar 5, 2013 #1
    1. The problem statement, all variables and given/known data

    Suppose that A and B are finite sets.
    What is |P(AxB)|? Meaning what is the cardinality of the power set of a cartesian product of the sets A and B.


    2. Relevant equations

    |AxB|=|A| * |B| since A and B are finite sets
    Power set of a set is the set of all subsets of that set, including the empty set and the set itself
    There are 2^|A| subsets for a set A when A is finite


    3. The attempt at a solution

    Since A and B are finite sets, we have |AxB|=|A| * |B|. Now the power set of (AxB) is the set of all its subsets, including the empty set and the set AxB itself. Since A and B are both finite sets, there is also a finite number of subsets of (AxB). By letting C=AxB, there are exactly 2^|C| subsets. Thus |P(AxB)|=2^|AxB|=2^(|A| * |B|)

    This is what I have.
     
  2. jcsd
  3. Mar 5, 2013 #2

    jbunniii

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    Looks fine to me.
     
  4. Mar 5, 2013 #3

    Dick

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    That's just fine.
     
  5. Mar 5, 2013 #4
    I have a similar question asking what the |P(A)xP(B)| is. Can I ask that in this thread, or do I have to start another thread?
     
  6. Mar 5, 2013 #5

    Dick

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    You never HAVE to start another thread. It's usually recommended if the problem is unrelated but this one isn't. Ask here.
     
  7. Mar 5, 2013 #6
    The question asks what is |P(A)xP(B)|. This is asking what is the cardinality of the cartesian product of the power set of A and power set of B. I'm having some trouble deciphering this.

    I let C=P(A) and D=P(B). Then, |CxD|=|C| * |D| since once again A and B are both finite sets, meaning their respective power sets, C and D, are also finite sets.

    So, |P(A)xP(B)|=2^C * 2^D

    That is what I have.
     
  8. Mar 5, 2013 #7

    Dick

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    No problem again. That's correct.
     
  9. Mar 5, 2013 #8
    Dick, are you saying that the syntax or form looks correct, or do you believe that is the correct answer?

    So simplyifying it, it would be |P(A)xP(B)|=2^|C| * 2^|D|=2^2^|A| * 2^2^|B|
     
  10. Mar 5, 2013 #9

    Dick

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    No, I was reading too fast. It's 2^|A|*2^|B|. |C|=2^|A| and |D|=2^|B|. |CxD|=|C||D|.
     
    Last edited: Mar 5, 2013
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