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.

$\blacksquare$

Sources

• Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.2$