Let {A1,A2...} be a sequence of events of a sample...?

Question

Let {A1,A2,...} be a sequence of events of a sample space S. Find a sequence {B1,B2,...} of mutually exclusive events such that for all n>=1,

U(from i=1 to n) Ai = U(from i=1 to n) Bi.

I'm thinking that the events B1... must be the parts that are mutually exclusive in A1...

Answer 1

OOOH! ... I just realized what it meant.

Here's the idea:

Let X - Y be the complement of Y in X... that is, X - Y = X ∩ Y'.  (You might be representing it as X \ Y... that works fine too)

B1 = A1

B2 = A2 - B1

B3 = A3 - B2

...

in general:

Bk = Ak - B(k-1).

Then all the events Bk are mutually exclusive among one another.  The idea being that we only need to add elements to the union that weren't there before.