Results 1 to 4 of 4

Thread: intersection of union

  1. #1
    Member
    Joined
    Dec 2008
    Posts
    154

    intersection of union

    i do not understand how this notation works. $\displaystyle \bigcap_{n=1}^\infty$$\displaystyle \bigcup_{m \geq n}$$\displaystyle A_m$
    help me please.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Aug 2009
    Posts
    125
    Hi, i don't know if the following visualisation is helpful:
    $\displaystyle \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m = \bigcap_{n=1}^\infty (A_n \cup A_{n+1}\cup A_{n+2}\cup A_{n+3}\ldots) =$

    $\displaystyle = (A_1 \cup A_{2}\cup A_{3}\cup A_{4}\cup A_{5}\cup A_{6}\ldots)\cap (A_{2}\cup A_{3}\cup A_{4}\cup A_{5}\cup A_{6}\ldots)\cap (A_{3}\cup A_{4}\cup A_{5}\cup A_{6}\ldots)\cap\ldots$

    but maybe this little proposition will make things clear:

    $\displaystyle x \in \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m \mbox{ if and only if } x \in A_k \mbox{ for infinitely many } k \in \mathbb{N}.$

    Proof: "=>" assume $\displaystyle x\in A_k$ for only finitely many $\displaystyle k\in\mathbb{N}$, let $\displaystyle l$ be the biggest such $\displaystyle k$, then for all $\displaystyle m\ge l+1$ we have $\displaystyle x\not \in A_m$. This means $\displaystyle x \not \in \bigcup_{m\ge l+1}A_m$, so $\displaystyle x\not\in \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m$.

    "<=" assume $\displaystyle x\not\in \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m$, this means there's some $\displaystyle n\in\mathbb{N}$ such that $\displaystyle x \not \in \bigcup_{m\ge n}A_m$, that is, $\displaystyle x\not\in A_m$ for all $\displaystyle m\ge n$, so $\displaystyle x$ belongs to only finitely many $\displaystyle A_k$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Dec 2008
    Posts
    154
    So if we define $\displaystyle lim_{n \rightarrow \infty}sup$ $\displaystyle A_m$=$\displaystyle \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m$, what is $\displaystyle sup$ $\displaystyle
    A_5$?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Aug 2009
    Posts
    125
    Quote Originally Posted by Kat-M View Post
    So if we define $\displaystyle lim_{n \rightarrow \infty}sup$ $\displaystyle A_m$=$\displaystyle \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m$, what is $\displaystyle sup$ $\displaystyle
    A_5$?
    this doesn't make sense.
    it is "limes superior of a sequence of sets $\displaystyle \{A_n:\, n\in \mathbb{N}\}$" , by definition:
    $\displaystyle \limsup_{n\rightarrow \infty}A_n =\bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Union and intersection
    Posted in the Geometry Forum
    Replies: 6
    Last Post: May 18th 2011, 03:17 PM
  2. intersection and union...
    Posted in the Statistics Forum
    Replies: 2
    Last Post: Aug 31st 2009, 07:43 PM
  3. Union & Intersection
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: Jul 16th 2009, 06:54 AM
  4. Union (u) and Intersection (n)
    Posted in the Algebra Forum
    Replies: 6
    Last Post: Jan 17th 2009, 06:39 AM
  5. intersection and union
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Oct 8th 2007, 04:24 PM

Search Tags


/mathhelpforum @mathhelpforum