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Math Help - Show that power set is a sigma-algebra

  1. #1
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    Show that power set is a sigma-algebra

    Hi

    Let  \Omega be a finite set. Show that the set of all subsets of  \Omega , 2^{\Omega}, is also finite and that it is a \sigma-algebra.

    Solution (?)

    Since  \Omega is finite,  \exists  n \in \mathbb{N} such that card(\Omega) \leq n . From set theory we know that the number of subsets of a finite set of cardinality  n is 2^{n}. Therefore,
    card(2^{\Omega}) \leq 2^{n} < \infty , since  n < \infty . Hence, 2^{\Omega} is finite.
     \left\{\emptyset\right\} and \left\{\Omega\right\} is in 2^{\Omega}. Since 2^{\Omega} contains all subsets of  \Omega it must also contain all necessary unions, intersections and complements to make 2^{\Omega} a \sigma-algebra.

    Comments/improvements?

    Thanks!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by ecnanif View Post
     \left\{\emptyset\right\} and \left\{\Omega\right\} is in 2^{\Omega}
    You should say \emptyset,\;\Omega instead of \left\{{\emptyset}\right\},\;\left\{{\Omega}\right  \}.

    Regards.

    Fernando Revilla
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  3. #3
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    Quote Originally Posted by FernandoRevilla View Post
    You should say \emptyset,\;\Omega instead of \left\{{\emptyset}\right\},\;\left\{{\Omega}\right  \}.

    Regards.

    Fernando Revilla
    Ok, but besides from this, correct?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by ecnanif View Post
    Ok, but besides from this, correct?
    Yes.

    Regards.

    Fernando Revilla
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