A question about Sequence Space

[luckyman]

1. Show that Ao, the collection of all cylinders of all rank is a field.

A cylinder of rank n is a set of the form { w∈S^∞ : R1(w)R2(w)...Rn(w) ∈ H}
where H is a set of n-long sequences of elements of S. That is H is a subset of S^n

Example:
now think about a toss a coin question.
0(tail=fail) and 1(head = success)
here Ai is the event that ith toss is a head
A1= { w: R1(w)∈{1}}
A2= { w: R1(w)R2(w)∈{11,01}
A3= { w: R1(w)R2(w)R3(w)∈{111,101,011,001}
.
.
.so all the Ai are cylinder sets.

Now my question is let Ao be the collection of all cylinders of all rank, then is Ao a field ?

 

[Mark44]

You're not likely to get a reply other than this one if you don't show some attempt at your problem.

 

[luckyman]

I changed my question with more details.
I think to show this collection is a field, I have to show the complements of each elements is again in this set and the union of each element is also in this set. But I think this properties are for sigma field. Are they valid also for field ?

Or to show just that

A and B be two elements of this set then I have to show that A union B is also in this set and
when A is an element of this set its complement is also in this set

is enough for field ?