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Thread: Limit of sequence of sets

  1. #1
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    Limit of sequence of sets

    I am trying to prove a proposition, but it's proving harder than I expected. I was wondering if someone could lead me in the right direction. Please don't give full answers. I'm just looking for a hint. The problem says:
    If $\displaystyle \{E_n\}$ is a sequence of sets and $\displaystyle D_1=E_1$, $D_{n+1}=D_n\bigtriangleup E_{n+1}, n=1,2,...$, show that $\displaystyle lim D_n$ existis if and only if $\displaystyle lim E_n = \emptyset$.

    Thank you.
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  2. #2
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    Re: Limit of sequence of sets

    I don't know exactly what "the limit of a sequence of sets" has been defined to mean in your problem. However, I can show you something that I think might prove this claim for your whatever your definition there is. I'll just state what I can prove, and then leave it to you to see if that, when considered in light of whatever definition you have of "convergence sequences of sets", proves this claim. I suspect this will do the job.
    (And it appears my suspicion proved wrong. Typical. )

    Given a set $\displaystyle \mathbb{X}$ and $\displaystyle \mathcal{E} = \{E_n\}_1^{\infty} \subset \mathcal{P}(\mathbb{X})$, define $\displaystyle D_1=E_1$, and then inductively $\displaystyle D_{n+1}=D_n\bigtriangleup E_{n+1}, n=1,2,...$.

    Let $\displaystyle E^* = \bigcap_{n = 1}^{\infty}E_n$.

    Claim #1: If $\displaystyle E^* \ne \emptyset$ and $\displaystyle x \in E^*$, then $\displaystyle x \in D_n \Leftrightarrow n$ is odd.

    Proof: Will be by induction.

    Note that $\displaystyle x \in E^* \Rightarrow x \in E_n \ \forall n \in \mathbb{N}$.

    Now let $\displaystyle S(m)$ be the proposition "$\displaystyle k \in \mathbb{N} \ni 1 \le k \le 2m \Rightarrow (x \in D_k \Leftrightarrow k$ is odd)".

    Since $\displaystyle x \in E_1$, have that $\displaystyle x \in D_1 = E_1$. But also $\displaystyle x \in E_2$, so it follows that $\displaystyle x \notin D_2 = D_1 \bigtriangleup E_2$.

    Thus have proven that $\displaystyle x \in D_1$, and $\displaystyle x \notin D_2$. That proves that $\displaystyle S(1)$ is true.

    Assume $\displaystyle S(n)$ is true for some $\displaystyle n \in \mathbb{N}, n \ge 1$.

    Let $\displaystyle k=2n$. Then $\displaystyle k$ is even, and $\displaystyle k \le 2n$. Thus $\displaystyle S(n)$ true implies that $\displaystyle x \notin D_{k} ( = D_{2n})$.

    So $\displaystyle x \notin D_{2n}$, but also $\displaystyle x \in E_{2n+1}$. Thus $\displaystyle x \in D_{2n+1}=D_{2n} \bigtriangleup E_{n+1}$.

    Now $\displaystyle x \in D_{2n+1}$, but also $\displaystyle x \in E_{2n+2}$. Thus $\displaystyle x \notin D_{2n+2}=D_{2n+1} \bigtriangleup E_{n+1}$.

    Have shown that $\displaystyle S(n)$ true implies that $\displaystyle x \in D_{2n+1}$ and also that $\displaystyle x \notin D_{2n+2}$.

    But those three statements ($\displaystyle S(n)$ true, $\displaystyle x \in D_{2n+1}$, and $\displaystyle x \notin D_{2n+2}$) together exactly comprise the statement $\displaystyle S(n+1)$.

    Thus $\displaystyle S(n) \Rightarrow S(n+1) \ \forall n \in \mathbb{N}$. Also have shown that $\displaystyle S(1)$ is true.

    Thus by induction, have proven $\displaystyle S(n)$ is true $\displaystyle \forall n \in \mathbb{N}$. That proves Claim #1.

    Claim #2: If $\displaystyle x \in \mathbb{X}$ and there exists $\displaystyle N \in \mathbb{N}$ such that $\displaystyle x \notin E_n \ \forall \ n > N$ (always here $\displaystyle n \in \mathbb{N}$),

    then $\displaystyle x \in D_N \Rightarrow x \in D_n \ \forall \ n \ge N$,

    and $\displaystyle x \notin D_N \Rightarrow x \notin D_n \ \forall \ n \ge N$.

    Proof:

    If $\displaystyle x \in D_N$, then since $\displaystyle x \notin E_{N+1}$, have that $\displaystyle x \in D_{N+1}=D_N\bigtriangleup E_{N+1}$.

    Next, since $\displaystyle x \in D_{N+1}$ and $\displaystyle x \notin E_{N+2}$, can conclude that $\displaystyle x \in D_{N+2}=D_{N+1}\bigtriangleup E_{N+2}$.

    By the obvious induction, $\displaystyle x \in D_n \ \forall n \ge N$.

    Conversely, if $\displaystyle x \notin D_N$, then $\displaystyle x \notin E_{N+1} \Rightarrow x \notin D_{N+1}=D_N\bigtriangleup E_{N+1}$.

    The induction is clear: for $\displaystyle n \ge N, x$ will be in neither $\displaystyle E_{n+1}$ (by the definition of $\displaystyle N$), nor $\displaystyle D_n$ (by induction), and therefore $\displaystyle x$ won't be in $\displaystyle D_{n+1}$.

    That proves Claim #2.
    Last edited by johnsomeone; Sep 18th 2012 at 02:10 PM.
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  3. #3
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    Re: Limit of sequence of sets

    Quote Originally Posted by johnsomeone View Post
    I don't know exactly what "the limit of a sequence of sets" has been defined to mean in your problem. However, I can show you something that I think might prove this claim for your whatever your definition there is. I'll just state what I can prove, and then leave it to you to see if that, when considered in light of whatever definition you have of "convergence sequences of sets", proves this claim. I suspect this will do the job.

    Given a set $\displaystyle \mathbb{X}$ and $\displaystyle \mathcal{E} = \{E_n\}_1^{\infty} \subset \mathcal{P}(\mathbb{X})$, define $\displaystyle D_1=E_1$, and then inductively $\displaystyle D_{n+1}=D_n\bigtriangleup E_{n+1}, n=1,2,...$.

    Let $\displaystyle E^* = \bigcap_{n = 1}^{\infty}E_n$.

    Claim #1: If $\displaystyle E^* \ne \emptyset$ and $\displaystyle x \in E^*$, then $\displaystyle x \in D_n \Leftrightarrow n$ is odd.

    Proof: Will be by induction.

    Note that $\displaystyle x \in E^* \Rightarrow x \in E_n \ \forall n \in \mathbb{N}$.

    Now let $\displaystyle S(m)$ be the proposition "$\displaystyle k \in \mathbb{N} \ni 1 \le k \le 2m \Rightarrow (x \in D_k \Leftrightarrow k$ is odd)".

    Since $\displaystyle x \in E_1$, have that $\displaystyle x \in D_1 = E_1$. But also $\displaystyle x \in E_2$, so it follows that $\displaystyle x \notin D_2 = D_1 \bigtriangleup E_2$.

    Thus have proven that $\displaystyle x \in D_1$, and $\displaystyle x \notin D_2$. That proves that $\displaystyle S(1)$ is true.

    Assume $\displaystyle S(n)$ is true for some $\displaystyle n \in \mathbb{N}, n \ge 1$.

    Let $\displaystyle k=2n$. Then $\displaystyle k$ is even, and $\displaystyle k \le 2n$. Thus $\displaystyle S(n)$ true implies that $\displaystyle x \notin D_{k} ( = D_{2n})$.

    So $\displaystyle x \notin D_{2n}$, but also $\displaystyle x \in E_{2n+1}$. Thus $\displaystyle x \in D_{2n+1}=D_{2n} \bigtriangleup E_{n+1}$.

    Now $\displaystyle x \in D_{2n+1}$, but also $\displaystyle x \in E_{2n+2}$. Thus $\displaystyle x \notin D_{2n+2}=D_{2n+1} \bigtriangleup E_{n+1}$.

    Have shown that $\displaystyle S(n)$ true implies that $\displaystyle x \in D_{2n+1}$ and also that $\displaystyle x \notin D_{2n+2}$.

    But theose three statements together exactly comprise the statement $\displaystyle S(n+1)$.

    Thus $\displaystyle S(n) \Rightarrow S(n+1) \ \forall n \in \mathbb{N}$. Also have shown that $\displaystyle S(1)$ is true.

    Thus by induction, have proven $\displaystyle S(n)$ is true $\displaystyle \forall n \in \mathbb{N}$. That proves the claim #1.

    Claim #2: If $\displaystyle x \in \mathbb{X} \ni \exists N \in \mathbb{N} \ni \ \forall n \in \mathbb{N}, n > N, x \notin E_n$,

    then $\displaystyle x \in D_N \Rightarrow x \in D_n \forall n \in \mathbb{N}, n > N$, and $\displaystyle x \notin D_N \Rightarrow x \notin D_n \forall n \in \mathbb{N}, n > N.$.

    Proof:

    If $\displaystyle x \in D_N$, then $\displaystyle x \notin E_{N+1} \Rightarrow x \in D_{N+1}=D_N\bigtriangleup E_{N+1}$.

    Then $\displaystyle x \in D_{N+1}$ and $\displaystyle x \notin E_{N+1}$ again together imply that $\displaystyle x \in D_{N+2}=D_{N+1}\bigtriangleup E_{N+2}$.

    By the obvious induction, $\displaystyle x \in D_n \ \forall n > N$.

    Conversely, if $\displaystyle x \notin D_N$, then $\displaystyle x \notin E_{N+1} \Rightarrow x \notin D_{N+1}=D_N\bigtriangleup E_{N+1}$.

    The induction is clear: for $\displaystyle n \ge N, x$ will be in neither $\displaystyle E_{n+1}$ (by the definition of $\displaystyle N$), nor $\displaystyle D_n$ (by induction), and therefore $\displaystyle x$ won't be in $\displaystyle D_{n+1}$.

    That proves Claim #2.
    Thanks for the response. I actually found a way of proving it using the characteristic function. It is quite a short proof.
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