A value function on a set \(A\) assigns 0 or 1 to each subset of \(A\).

1. If \(A\) has 3 elements, how many different value functions are there on \(A\)?

There are \(2^3 = 8\) subsets of \(A\). If each of them can be either 0 or 1, then there are \(2^{2^3} = 256\) different value functions on \(A\).

2. What if \(A\) has \(n\) elements?

\(2^{2^n}\)