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Question on sets

  1. May 15, 2012 #1
    Question is in paint doc. Determine if the statement is true or false.

    My solution:
    I have two solutions

    Sol 1: FalseIf A1 contains A2 and A2 contains A3 then the number of elements of A3 contained in A1 is less than the number of elements in A2 contained in A1. In other words the intersection of A1 and A3 has fewer elements than the intersection of A1 and A3. Therefore the intersection cannot be infinite if the elements in each consecutive intersection are decreasing.

    Sol 2: True

    If A1, A2,... An are sets of infinite number of elements then A1 = A2 = ... = An. For instance, how can a set containing all negative real numbers (-∞,0] and a set containing all positive real numbers [,+∞) be infinite if the set of all read numbers contains more elements than the two sets above? Therefore is it true if two or more sets have infinite elements then those sets are equal?
     

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  3. May 15, 2012 #2
    Both of your solutions are wrong :)

    False: The intersection can still be infinite

    True: of course it's not true that A1 = A2 = ... = An in general.

    Either you need to show that the intersection is always infinite, or find a counterexample.
     
  4. May 15, 2012 #3

    vela

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    Neither solution is correct. You're applying your intuition from dealing with finite sets to infinite sets. That doesn't work. You can remove an infinite number of elements from an infinite set and still have an infinite number of elements remaining.
     
  5. May 15, 2012 #4
    If A1 contains An and An does not contain A1,A2...An-1 and taking in consideration that An is a set of infinite elements the intersection of A1,A2,... and An will be An which is a set of infinite elements.

    Statement is True.
     
    Last edited: May 15, 2012
  6. May 15, 2012 #5

    vela

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    You're misreading the problem. It's asking you about the set
    $$A = \bigcap_{i=1}^\infty A_i,$$ the intersection of all of the sets. The intersection between any two of the sets will obviously contain an infinite number of elements, so it's not a very interesting question to ask.
     
  7. May 15, 2012 #6
    I understand the question. I was thinking the intersection of all the sets would be An.
    If you noticed I said "the intersection of A1,A2,... and An ... "
     
  8. May 15, 2012 #7
  9. May 15, 2012 #8

    vela

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    How can An be the intersection of all the sets? What about An+1?
     
  10. May 15, 2012 #9
    A simple way to show a counterexample works is if you can show that for every [itex]x \in A_1 [/itex] there exists a [itex] n \in \mathbb{N} [/itex] such that [itex] x \notin A_n [/itex]
     
  11. May 15, 2012 #10
    I've thought of that but I can't think of an example and I can't see how that is possible if An is a subset of A1
     
  12. May 15, 2012 #11
    Why would the intersection be empty? Am and Am+1 still have m+1, m+2, m+3,... in common, therefore the intersection of Am and Am+1 would be m+1, m+2, and so on... I must have the wrong interpretation of intersection.
     

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  13. May 15, 2012 #12

    vela

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    Why do you keep talking about the intersection of two sets? As I said earlier, the question is asking about the intersection of all of the sets, i.e. ##A = A_1 \cap A_2 \cap A_3 \cap \cdots ##
     
  14. May 15, 2012 #13
    Think again. Maybe start as easy as you can, and choose [itex]A_1 = \mathbb{N} [/itex]
     
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