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Cardinality of the set of all finite subsets of [0,1]

  1. Dec 12, 2011 #1
    Hello, I was wondering this, what is the cardinality of the set of all finite subsets of the real interval [0,1]

    It somehow confuses me because the interval is nonnumerable (cardinality of the continuos [itex] \mathfrak{c}[/itex]), while the subsets are less than numerable (finite). It is clear that it has to be equal or greater than [itex] \mathfrak{c}[/itex] because one can consider subsets of only one element and there you got one set for each real number in the interval. It is equal to it, isn't?

    Thanks.
     
  2. jcsd
  3. Dec 12, 2011 #2
    You are correct that the cardinality is at least as much as the cardinality of [0.1]. The power set of an infinite set always of greater cardinality than the base set. I dunno about the finite subsets.
     
  4. Dec 12, 2011 #3

    micromass

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    Yes. Equality holds. To see this, try to figure out the cardinality of

    [tex]\mathcal{A}_n=\{A\subseteq [0,1]~\vert~|A|=n\}[/tex]

    Then the set you are looking for is

    [tex]\bigcup_n \mathcal{A}_n[/tex]

    a countable union.
     
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