Few definitions

S = Universal set

A,B,C,D,.... = Subsets of S

A' = compliment of A

+ = union on sets

. = intersection of sets

Let me take a case when there are just two subsets A and B

Now, A.B, A'.B, A.B', A'.B' - partition S into mutually exclusive and exhaustive sets.

Now any 'valid' operation on A and B (using +,.,') will represent a set in S, say X

Then this X can be representes a union of some/all of A.B, A'.B, A.B', A'.B'

For e.g A+B = A.B + A'.B + A.B'

X can be as complicated as you want. (by repeatedly applying these operators in a consistent way)

Now I have two questions

1. How do you prove the above result - in a most generalized form?

2. Will the proof / result change if

a> S has infinite elements

b> S has infinite subsets

I do have an idea but am getting mixed up. Any help/pointers plz?