Union or intersection of several infinite sets

[hahamonkey]

Just started on the 'Introduction to Probability textbook by John and Dimitri and it hit me with this notation description.



I am struggling to understand what it means/how to decode it and would appreciate an implemented example. The 'for some n' and 'for all n' part is confusing.

My attempt to understand this:
For every positive integer n 1,2,3...
we are given set n...meaning S1={1}, S2={2}, S3={3}...

If it's the union of all sets, isn't it Sn = {1,2,3...},? what does the for some n mean?
If it's the intersection of all sets, isn't it Sn = empty? what does the for all n mean?

 

[Harry_the_cat]

Let's just say that A ={1, 2, 3} and B = {2, 3, 4}.
The union is {1, 2, 3, 4, 5} ie x is an element of at least one of the sets (some)
The intersection is {2, 3} ie x is an element of both of the sets (all).

S1 does not necessarily equal {1}. It is just a name given to one of the sets, just like A above.

 

[hahamonkey]

Oh I see, it seems like I misunderstood it as infinitely many n. Thanks!

 

[Harry_the_cat]

Oh I see, it seems like I misunderstood it as infinitely many n. Thanks!

The same would apply for anywhere from 1, 2, 3, 4, ... sets.