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Math Help - Sequences of Sets and Symmetric Difference

  1. #1
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    Sequences of Sets and Symmetric Difference

    Hi everyone, I have a question about some properties of limit of sequence of sets. Here we go:
    Let \{E_n\} and \{D_n\} be two sequences of sets such that they both converge. That is, \varlimsup E_n=\varliminf E_n=E and \varlimsup D_n=\varliminf D_n=D. Can we conclude any of the following?
    i. \lim (E_n \cup D_n) = E \cup D
    ii. \lim (E_n \cap D_n) = E \cap D
    iii. \lim (E_n - D_n) = E - D
    iv.  \lim (E_n \Delta D_n) = E \Delta D where \Delta denotes symmetric difference

    Thanks for your help.
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  2. #2
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    Re: Sequences of Sets and Symmetric Difference

    Quote Originally Posted by sepmo View Post
    Hi everyone, I have a question about some properties of limit of sequence of sets. Here we go:
    Let \{E_n\} and \{D_n\} be two sequences of sets such that they both converge. That is, \varlimsup E_n=\varliminf E_n=E and \varlimsup D_n=\varliminf D_n=D.
    Can you tell us what you mean by a "limit of sequence of sets"?
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  3. #3
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    Re: Sequences of Sets and Symmetric Difference

    Thanks for your interest in my question. Let \{E_n\} be a sequence of sets in the class \mathcal{A}. Then, we define the lower limit and the upper limit of this sequence as \varliminf E_n=\cup_{n=1}^\infty \cap_{m=n}^\infty E_m and \varlimsup E_n=\cap_{n=1}^\infty \cup_{m=n}^\infty E_m respectively. We say that \lim E_n exists if \varliminf E_n = \varlimsup E_n and is equal to the lower and the upper limits.
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  4. #4
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    Re: Sequences of Sets and Symmetric Difference

    If A is a set, let in(x, A) = 1 if x is in A and in(x, A) = 0 otherwise. Then lim (A_n) exists iff for every x, the limit of in(x, A_n) exists, i.e., in(x, A_n) eventually stabilizes. Also, x is in lim(A_n) iff in(x, A_n) stabilizes at 1. It is easy to express in(x, A * B) through in(x, A) and in(x, B) where * is each of the four set operations in the question. Using these facts, it is possible to show that the limits in the left-hand sides exist and the equalities hold.
    Thanks from sepmo
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    Re: Sequences of Sets and Symmetric Difference

    Quote Originally Posted by emakarov View Post
    If A is a set, let in(x, A) = 1 if x is in A and in(x, A) = 0 otherwise. Then lim (A_n) exists iff for every x, the limit of in(x, A_n) exists, i.e., in(x, A_n) eventually stabilizes. Also, x is in lim(A_n) iff in(x, A_n) stabilizes at 1. It is easy to express in(x, A * B) through in(x, A) and in(x, B) where * is each of the four set operations in the question. Using these facts, it is possible to show that the limits in the left-hand sides exist and the equalities hold.
    I'm not quite sure what you mean by in(x, A). Is that an inner product?
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  6. #6
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    Re: Sequences of Sets and Symmetric Difference

    I am defining a function called "in".
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    Re: Sequences of Sets and Symmetric Difference

    Quote Originally Posted by emakarov View Post
    I am defining a function called "in".
    I see what you're saying. I'm going to think about it a bit more, and if I'm not convinced, I'll bug you again. Thanks for the help.
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