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A question for order in sets

  1. May 22, 2008 #1
    I am reading a book named sets & groups,
    I would like to ask:
    What exactly is a finite set of order n? [itex]A_1\times A_2\times A_3...A_n[/itex] ?
    is pair [itex](x_1,x_2,x_3...x_m)[/itex] counted as a single element? or a set? or an order?
    also how should I prove that the number of subsets, including empty set, of a finite set of order n is [itex]2^n[/itex]?


    Thanks!
     
  2. jcsd
  3. May 22, 2008 #2

    CompuChip

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    The order of a set is just the number of elements in a set. So a finite set of order n is a set with n elements.

    You can also look at Cartesian products of set, and then the elements of that set will be pairs / triples / (n-)tuples of elements from those sets. You can prove that the order of the product [itex]A = A_1 \times A_2 \times \cdots \times A_n[/itex] is [itex]|A| = |A_1| \times |A_2| \times \cdots \times |A_n|[/itex] where |X| denotes the order of set X.

    For your last question: note that if you have finitely many elements you can list them all. One way to proceed is to write down a one or zero for each of them, indicating whether they are in a given subset or not. You can put all of these in an n-tuple. Then show that each such n-tuple corresponds to one particular subset and that each subset can be written as one particular n-tuple; finally count then number of n-tuples you can make.
     
  4. May 22, 2008 #3

    HallsofIvy

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    A set containing exactly n members.

     
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