Well I would say no becuase in the way you enumerated

How would you define

for

Sorry if this is getting more and more confusing - seems I am not able to convey the crux of the problem.

This is what it is

1. Let there be any universal set S

2. Let there be any subsets of this universal set, we call them

3. Based on

we define partitions (as in my post above). In general there will be

such partitions, some of which may be

(null set)

4. Now let X be any set whcih can be constructed from

using the operators below

a> Intersection

b> Union

c> Complement

5. Prove that X can be represented as union of partitions (as we defined them in step 3 above)

Also I would write down how I think we can proceed with the proof.

Show that if there are any two subsets of S, A and B where A,B can be expressed as union of the partitions then each of the following can also be expressed as union of paritions

1. A' (or B')

2.

3.

Now to start with each of the subsets

can be expressed as sum of partitions hence any X can also be.

What I am struggling with is the case when we make 'n' infinite. Not sure if my logic breaks down somewhere?