By 2^S it's not meant to raise 2 to a power (although there's a reason that notation is used), but rather to take all possible subsets of S. Then you partition 2^S (the set of all subsets) into equivalence classes based on their cardinality. So the class of subsets with zero elements is {{}}, the class of one-element subsets is {{1},{2},{3},{4}}, and so forth.

How many subsets have four elements?