I am confused, is n the cardinality of every set in S1 or is n the cardinality of S1? I read it as the first meaning but it looks like you interpreted it as the second.

Not every member of S2 is a member of S1.

Suppose S1={T1,T2}

The the subsets of S1 are {T1}, {T2}, {T1,T2}, {} where {} is the empty set.

{T1,T2} is not a member of S1 and nor is {}