Understanding the Limsup Operation in Set Theory

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Discussion Overview

The discussion revolves around the lim sup operation in set theory, specifically the expression lim sup_n A_n = ∩^∞ U^∞ A_k. Participants seek clarification on the meaning of this operation, its components, and the order of operations involved.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant expresses confusion about the lim sup operation and asks whether it involves taking a union followed by an intersection.
  • Another participant inquires about the meaning of combining intersection and union in this context.
  • A participant explains that the order of operations is important, similar to evaluating multiplication and summation, suggesting that operations should be performed from right to left unless brackets indicate otherwise.
  • Some participants emphasize the need to specify limits in the expression for lim sup, providing the formulation limsup = ∩(n=1,∞)U(k=n,∞)A_k.
  • One participant breaks down the lim sup operation into two steps: defining B_n as the union of sets starting from index n, followed by the intersection of all B_n, indicating that it captures points present in infinitely many A_k.
  • A later reply acknowledges the clarity of this explanation, indicating it aligns with the original question posed.

Areas of Agreement / Disagreement

Participants generally agree on the steps involved in understanding the lim sup operation, but there is still some confusion regarding the interpretation of the expression and the order of operations.

Contextual Notes

Participants have not resolved all aspects of the lim sup operation, particularly regarding the implications of the right-hand side of the expression and the specific limits involved.

woundedtiger4
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Sorry if the title of my question is wrong,

lim sup_n A_n = ∩^∞ U^∞ A_k

does it mean that first we are taking the union of A_n and then we are taking intersection? actually I am confused, what is it actually?
 
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when we write intersection & union together then what does it mean?
 
Hey woundedtiger4.

Usually if we do this it depends on the order that it's written in much in the same way that if you have pi symbol (multiplication) and a sigma symbol (summation), you evaluate it from right to left if there are no brackets.

So if you have pi sigma (expression) then you calculate the sigma terms first before doing the pi terms on the relevant indices.
 
You need to specify the limits. limsup = ∩(n=1,∞)U(k=n,∞)A_k
 
mathman said:
You need to specify the limits. limsup = ∩(n=1,∞)U(k=n,∞)A_k
yeah that exact expression but what does mean by the right hand side?
 
woundedtiger4 said:
yeah that exact expression but what does mean by the right hand side?

First step: Let B_n = ∪(k=n,∞)A_k, that is the union of all sets with indices starting at n.

Second step: Limsup = ∩(n=1,∞)B_n, that is the intersection of all B_n.

In essence you are getting all the points which are in an infinite number of A_k.
 
mathman said:
First step: Let B_n = ∪(k=n,∞)A_k, that is the union of all sets with indices starting at n.

Second step: Limsup = ∩(n=1,∞)B_n, that is the intersection of all B_n.

In essence you are getting all the points which are in an infinite number of A_k.

excellent explanation... this is what I asked in my main question...

Thanks a lot.
 

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