The notation \\(2^S\\) is also consistent with the more general exponential notation \\(B^A\\) for sets \\(A\\), \\(B\\). The expression \\(B^A\\) denotes the set of all functions from A to B.

So according to that, \\(2^S\\) would be these set of all functions from \\(S\\) into 2. Here, we'll take '2' to be a name for two-element set {0,1}.

What's a function from \\(S\\) into {0,1}? It can be viewed as a predicate, which assigns true or false to every member of S.

And that's equivalent to a subset of S -- the subset of all values in S that map to 1.

So, as a set of functions, \\(2^S\\) is equivalent to \\(2^S\\) as a set of subsets.

So according to that, \\(2^S\\) would be these set of all functions from \\(S\\) into 2. Here, we'll take '2' to be a name for two-element set {0,1}.

What's a function from \\(S\\) into {0,1}? It can be viewed as a predicate, which assigns true or false to every member of S.

And that's equivalent to a subset of S -- the subset of all values in S that map to 1.

So, as a set of functions, \\(2^S\\) is equivalent to \\(2^S\\) as a set of subsets.