Understanding the Limsup Operation in Set Theory

[woundedtiger4]

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?

 

[woundedtiger4]

when we write intersection & union together then what does it mean?

 

[chiro]

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.

 

[mathman]

You need to specify the limits. limsup = ∩(n=1,∞)U(k=n,∞)A_k

 

[woundedtiger4]

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?

 

[mathman]

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.

 

[woundedtiger4]

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.