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Homework Help: How many elements are in two power sets

  1. Mar 30, 2008 #1
    [SOLVED] how many elements are in two power sets

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

    a.How many elements are in the power set {1,2,3,4}? b.How many elements are in the power set {1,2,3,4,5}?

    2. Relevant equations



    3. The attempt at a solution
    a. {empty set, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,3,4}} Are there 13 elements?

    b. {empty set, {1}, {2}, {3}, {4}, {5}, {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5}, {1,2,3}, {1,2,3,4}}

    Are there 19 elements?

    Is there a general rule as to how many elements are in a power set?

    Thank you very much
     
  2. jcsd
  3. Mar 30, 2008 #2

    Shooting Star

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    Homework Helper

    What happened to {2,3,4} and a few more?

    No. Like the last one, some are mising.

    It will be just the total number of combinations of 'n' things taken any number at a time, and one more (when you don't select any). Do you know how to find that?
     
  4. Mar 30, 2008 #3
    I got 16 elements in the first one. Does that look right?

    (It will be just the total number of combinations of 'n' things taken any number at a time, and one more (when you don't select any). Do you know how to find that?)

    No, I don't know how to do that. Could you please show me?

    Thank you
     
  5. Mar 30, 2008 #4

    Shooting Star

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    That is correct.

    Consider the first problem. You can choose 1, or not choose 1. So, 1 can be dealt with in two ways. For each of these cases, you can choose, or not choose 2. So, total number of ways to select either 1 or 2 or both or none is 2X2=4=2^2. Choosing none means you get the null set. Similarly, for the four elements 1 2 3 4, total number of choosing would be 2^4. All these selections is giving you the elements of the power set. So, the number of elements in the power set of the set {1,2,3,4} is 2^4.

    I'm sure you can tackle the next problem using this formula for the number of elements in the power set of a finite set.
     
  6. Mar 30, 2008 #5
    Thank you very much

    Regards
     
  7. Mar 30, 2008 #6
    Just writitng in general what actually Shooting Star already said and elaborated.

    If X is a set with |X|=n elements then: [tex]|P(X)|=2^{|X|}=2^{n}[/tex] so the power set has 2^n elements.
     
  8. Mar 30, 2008 #7
    Thank you very much

    Regards
     
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