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Set theoretic problem, lim sup and lim inf

  1. Apr 27, 2012 #1
    1. The problem statement, all variables and given/known data

    Given a sequence of sets {An}, n=1,2,...,∞, the lim sup [itex]S=\cap_{n=1}^\infty\cup^\infty_{k=n}A_k[/itex], and the lim inf [itex]I=\cup_{n=1}^\infty\cap^\infty_{k=n}A_k[/itex], obviously [itex]I\subset S[/itex], find an expression for the set difference [itex]S-I[/itex]

    2. Relevant equations

    [itex](A \cup B) \cap C = (A \cap C)\cup(B\cap C)\,[/itex]
    [itex] (A \cap B) \cup C = (A \cup C)\cap(B\cup C)\,[/itex]

    3. The attempt at a solution
    I don't know how to use the set algebra identities to simplify, I need to write S-I in terms of set differences of An. Are there any identities involving infinite union/intersection that might be of help? Thank you.
     
    Last edited: Apr 27, 2012
  2. jcsd
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