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Intersection of Indexed Sets 
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#1
Feb2211, 08:53 PM

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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
Feb2211, 09:07 PM

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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.



#3
Feb2311, 12:20 AM

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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.
Since x ∈ A_{i} for all i ∈ I, then clearly, x ∈ A_{1}, 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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