Given A1 superset of A2 superset of A3 superset of A4 .... and so on
how can i construct sets B1, B2, ...
so that each Bi's are disjoint.
The goal is to get
the infinite intersection of Ai = the infinite union of Bi
De morgans law:
(AUB)^c = (A^c N B^c)
(ANB)^c = (A^c U B^c)
which can be applied to infinite unions and intersections
The Attempt at a Solution
somewhere along the way, i understand that i need to involve de morgans law to turn the intersection into the union, but each Bi that I try gives me something strange and i can't come up with the result i want
The book suggests to use
B1 = A1
B2 = A2\B1
Bk = Ak\Bk-1
All the Bk's are disjoint, but...
what exactly is B2=A2\A1... i mean everything in A2, is technically in A1,
since A2 subset of A1. So an element in B2 is an element in A2, yet is not an element in A1, which then implies it is not an element of A2...
so i'm totally lost