Sequences of Sets and Symmetric Difference

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August 27th 2012, 06:52 AM
sepmo

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.
August 27th 2012, 07:42 AM
Plato

Re: Sequences of Sets and Symmetric Difference

Quote:

Originally Posted by sepmo

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"?
August 27th 2012, 08:00 AM
sepmo

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.
August 27th 2012, 10:58 AM
emakarov

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.
August 27th 2012, 11:14 AM
sepmo

Re: Sequences of Sets and Symmetric Difference

Quote:

Originally Posted by emakarov

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?
August 27th 2012, 11:24 AM
emakarov

Re: Sequences of Sets and Symmetric Difference

I am defining a function called "in".
August 27th 2012, 11:43 AM
sepmo

Re: Sequences of Sets and Symmetric Difference

Quote:

Originally Posted by emakarov

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. (Wink) Thanks for the help.