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.