Originally Posted by

**aman_cc** Sorry if my question was not clear. I'm not asking what you have understood.

Let me repeat it

Definitions -

S = Universal set

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

A' = compliment of A

+ = union on sets

. = intersection of sets

(this notation makes it easier to write the problem below)

Let there be 'n' subsets of S - A1, A2, A3, ...., An (any 'n' subsets will do)

Consider 2^n subsets -

A1.A2.A3...An-1.An

A1.A2.A3....An-1.An'

....

..

These 2^n subsets partition S into mutually exclusive and exhaustive sets. (This is easy to prove)

Now consider operations -

1. Union, +

2. Intersection, .

3. Compliment, '

Using the above operations and {A1,A2,A3,....,An} sets we can construct a new set, for e.g.

X = ((A1+A2)'.A3)+A7

**Prove that any such, X, can be represented as union of (some or all of the 2^n) partitions we constructed above.**

Will this proof change if

a> S has infinite elements

b> S has infinite subsets