Minimal Representation of a Set

Suppose there is a set T={e1, e2, e3, ..., e_n}, and some subsets S1,
S3, S3, ..., S_m of T.

Now given another subset Q of T, I want you to represent Q in terms of
S1...S_m and (1) set union operations disallowing overlapping (the two
additive sets should not have common elements) ; (2) set union
operations allowing overlapping; (3) set union operations disallowing
overlapping, and A/B set subtraction operations requiring B be a subset
of A; (4) set union operations allowing overlapping, and A/B set
subtraction operations not requiring B be a subset of A.

Terms (S1, S2, ..., S_m) used in your representation should be as few
as possible.

Regards,
Yao Ziyuan

.