Count of Boolean Functions

Theorem
There are $2^{\left({2^k}\right)}$ distinct boolean functions on $k\!$ variables.

Proof
Let $f: \mathbb B^k \to \mathbb B$ be a boolean function.

The domain of $f$ has $2^k$ elements, from Cardinality of Cartesian Product.

The result follows from Cardinality of Set of All Mappings.