I have been thinking about this - do not know if there is any formal way to approcah this problem
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?