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Homework Help: A question about Sequence Space

  1. Oct 20, 2009 #1
    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

    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 ?
    Last edited: Oct 20, 2009
  2. jcsd
  3. Oct 20, 2009 #2


    Staff: Mentor

    You're not likely to get a reply other than this one if you don't show some attempt at your problem.
  4. Oct 20, 2009 #3
    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 ?
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