Nested set question

**Danneedshelp** · August 30th 2009, 05:25 PM

Q: Decide which statements are true or false. If false, give an example of where it falls short.

a) If $A_{1}\supseteq\\A_{2}\supseteq\\A_{3}\supseteq\\A_ {4}...$ are all sets containing an infinite number of elements, then the intersection $\bigcap_{n=1}^{\infty}=A_{n}$ is infinite as well.

b) If $A_{1}\supseteq\\A_{2}\supseteq\\A_{3}\supseteq\\A_ {4}...$ are all sets containing an finite number of elements, then the intersection $\bigcap_{n=1}^{\infty}=A_{n}$ is finite and nonempty.

A:

a) True. I figure this is true, because every infinite interval on the real line can be broken down into a smaller interval; so, all the sets will tend towards some kind of singularity, but this will also be infinite.

b) False: suppose some element $m$ is in the intersection, we are saying this element must be in every single $A_{n}$ . This is clearly not true for $m+1\in\\\mathbb{N}$ . This is assuming $A_{n}=\{n,n+1,n+2,...\}$ .

Are my answers correct?

Thank you!

**Grandad** · August 30th 2009, 11:52 PM

Hello Danneedshelp

Originally Posted by Danneedshelp

Q: Decide which statements are true or false. If false, give an example of where it falls short.

a) If $A_{1}\supseteq\\A_{2}\supseteq\\A_{3}\supseteq\\A_ {4}...$ are all sets containing an infinite number of elements, then the intersection $\bigcap_{n=1}^{\infty}=A_{n}$ is infinite as well.

b) If $A_{1}\supseteq\\A_{2}\supseteq\\A_{3}\supseteq\\A_ {4}...$ are all sets containing an finite number of elements, then the intersection $\bigcap_{n=1}^{\infty}=A_{n}$ is finite and nonempty.

A:

a) True. I figure this is true, because every infinite interval on the real line can be broken down into a smaller interval; so, all the sets will tend towards some kind of singularity, but this will also be infinite.

b) False: suppose some element $m$ is in the intersection, we are saying this element must be in every single $A_{n}$ . This is clearly not true for $m+1\in\\\mathbb{N}$ . This is assuming $A_{n}=\{n,n+1,n+2,...\}$ .

Are my answers correct?

Thank you!

I'm afraid I don't understand your answer to (b). Your definition $A_{n}=\{n,n+1,n+2,...\}$ appears to define sets containing infinitely many elements.

The question as you have stated it doesn't insist that all the sets are non-empty. So if $\exists j, A_i = \{\}, i \ge j$ , then the intersection $\bigcap_{n=1}^{\infty}=\{\}$

But if all the sets are finite and non-empty, then isn't it the case that, after a certain point all the sets must be equal; i.e. $\exists j, A_i=A_j, i>j$ ? In which case, this 'limiting' set will be the intersection.

Grandad

**Plato** · August 31st 2009, 03:09 AM

Originally Posted by Danneedshelp

Q: Decide which statements are true or false. If false, give an example of where it falls short.

a) If $A_{1}\supseteq\\A_{2}\supseteq\\A_{3}\supseteq\\A_ {4}...$ are all sets containing an infinite number of elements, then the intersection $\bigcap_{n=1}^{\infty}=A_{n}$ is infinite as well.

Part a is also false.
$A_n = \left( {0,\frac{1}{n}} \right)\; \Rightarrow \;\bigcap\limits_n {A_n } = \emptyset$

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