The notation \\(2^S\\) also agrees with the exponential notation \\(B^A\\) for general sets \\(A\\), \\(B\\), which is defined as the set of all functions from A to B.
In this context, let's take 2 to mean the binary set \\(\lbrace True,False \rbrace \\).
So \\(2^S\\) means all functions going from \\(S\\) into \\(\lbrace True,False\rbrace \\). Each of these is a predicate, assigning \\(True\\) or \\(False\\) to every member of \\(S\\). And each predicate is equivalent to the subset of values in \\(S\\) which get mapped to \\(True\\).
So as a set of functions \\(2^S\\) is equivalent to the power set \\(2^S\\).