Intersection of Indexed Sets

  1. 1. The problem statement, all variables and given/known data
    Show that the intersection of Ai (for all i in I = {1, 2, 3, ... n } = A1. Ai is a subset of Aj whenever i <= j.


    2. Relevant equations



    3. The attempt at a solution
    Show:
    ***I'm having trouble showing part 1***1. that the intersection of Ai is a subset of A1, and
    2. A1 is a subset of the intersection of Ai.

    This is my attempt: 1. Let x be an element of the intersection of Ai. Then x is in Ai for all i in I. Since A1 is contained in all Ai, then x is contained in A1.

    2. Let x be an element of A1, then as A1 is a subset of Aj, for all j >= 1, x is an element of Aj. Thus, x is an element of the intersection of Ai.
     
  2. jcsd
  3. Dick

    Dick 25,887
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    I think that's completely correct. Except maybe that you don't need A1 is a subset of Ai for the first part. If x is in the intersection of the Ai, it's certainly in A1.
     
  4. SammyS

    SammyS 8,704
    Staff Emeritus
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    "This is my attempt: 1. Let x be an element of the intersection of Ai. Then x is in Ai for all i in I. [STRIKE]Since A1 is contained in all Ai, then x is contained in A1.[/STRIKE]"

    Since x ∈ Ai for all i ∈ I, then clearly, x ∈ A1, because 1 ∈ I .

    (Not that what you had was incorrect, but I think this is more direct.)

    You could do (2.) by induction.
     
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