An \(n\)-input, \(m\)-output boolean function is a function from \(\{TRUE, FALSE\}^n\) to \(\{TRUE, FALSE\}^m\). How many \(n\)-input, \(1\)-output boolean functions are there? How many \(n\)-input, \(m\)-output boolean functions are there?

First of all, these inputs are simply \(n\) length binary strings of which there are \(2^n\) possible forms. For single-output boolean functions (\(m=1\)) each possible input corresponds to 2 possible outputs, and so the total number of possible \(n\)-input, \(1\)-output boolean functions is \(2^{2^n}\).

Generalizing this approach for arbitrary values of \(m\) gives us \(2^m\) possible outputs and so the total number of \(n\)-input, \(m\)-output boolean functions is \(2^{m2^n}\).