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Intersection of a family of sets

  1. Aug 30, 2009 #1
    1. The problem statement, all variables and given/known data

    There is only a small issue that i am confused about... If we have a set


    [tex]\left(-\frac{1}{n},\frac{1}{n}\right)[/tex], where n is a natural number. If we want to find the intersection of all such sets, my question is whether the result will be the set containing only zrero,({0}), or the empty set?

    i.e

    [tex]\bigcap_{n=1}^{\infty}\left(-\frac{1}{n},\frac{1}{n}\right)=\{0\}...or...=\{\}??[/tex]

    If we apply the limit as n-->inf, it looks like it is gonna be the empty set, but i am not sure whether we can draw smilar parallels in this case?

    Suggestions?
     
  2. jcsd
  3. Aug 30, 2009 #2

    Hurkyl

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    Well, what is the definition of intersection?
    A number x is in the intersection if and only if it lies in each of the sets.​
    That is equivalent to the statement
    A number x is not in the intersection if and only if there exists a set it doesn't lie in.​

    ....
     
  4. Aug 30, 2009 #3

    Hurkyl

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    By the way, the interesction isn't a limit. (At least, it's not a limit in the sense you know it)

    Unlike addition which only operates on a pair of numbers at a time, and thus requires something extra to make sense of an idea like "infinite sum", intersection is naturally an infinitary operation -- it can be applied to any collection of sets, not just a pair of sets.
     
  5. Aug 30, 2009 #4
    Thnx for the reply Hurkyl.

    I know the def. of the intersection, but my issue was the inability to conclude whether 0 is an element of the set

    [tex]\left(-\frac{1}{n},\frac{1}{n}\right),\forall n \in N[/tex].


    If it were a closed set it would be part of it, but in this case i am having trouble figuring out whether it behaves similarly as when we take the limit, when n goes to infinity.

    ??
     
  6. Aug 30, 2009 #5

    Hurkyl

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    That isn't a set. That's a family of sets.

    Each particular value of n gives you a set. For which n is 0 in the set? For which n is 0 not in the set?
     
  7. Aug 30, 2009 #6
    Yes, my mistake.
    0 is in the set for every finite value of n... i am just not sure what happens when n-->infty. In other words, i am not sure how exactly the intersection of an indexed family of sets works, when the index set is not finite, in other words when it is infinite?
     
  8. Aug 30, 2009 #7
    good observation

    irrelevant, it is not a limit

    Read the answer from before:
    A number x is in the intersection if and only if it lies in each of the sets.

    So, for any natural number n, does 0 lie in (-1/n,1/n) ? Or is there some n for which 0 is outside of (-1/n,1/n) ?
     
  9. Aug 30, 2009 #8
    A quick comment. For every member of the family you have, n is finite. Since infinity is not an integer, there is no "infinity" member of the family.

    --Elucidus
     
  10. Aug 30, 2009 #9
    These are the key points i was debating with myelf about. I just thought whether there is any paralel drawn between a limit at infinty and the intersection of a family of sets when the index set is not finite.

    Thnx, i appreciate y'alls suggestions and help.

    Dilema resolved now.
     
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