Results 1 to 3 of 3

Math Help - Nested set question

  1. #1
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303

    Nested set question

    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!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1
    Hello Danneedshelp
    Quote Originally Posted by Danneedshelp View Post
    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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by Danneedshelp View Post
    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
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Nested sets
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: August 27th 2009, 10:11 PM
  2. Nested Quantifiers
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 23rd 2009, 03:41 PM
  3. nested interval thm in R^n
    Posted in the Calculus Forum
    Replies: 5
    Last Post: October 24th 2008, 09:00 AM
  4. Nested Quantifiers
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: October 6th 2008, 11:53 PM
  5. nested summation
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: September 20th 2008, 10:52 AM

Search Tags


/mathhelpforum @mathhelpforum