Count of Binary Operations on Set

Theorem
Let $S$ be a set whose cardinality is $n$.

The number $N$ of different binary operations that can be applied to $S$ is given by:


 * $N = n^{\paren {n^2} }$

Proof
A binary operation on $S$ is by definition a mapping from the cartesian product $S \times S$ to the set $S$.

Thus we are looking to evaluate:
 * $N = \card {\set {f: S \times S \to S} }$

The domain of $f$ has $n^2$ elements, from Cardinality of Cartesian Product of Finite Sets of Finite Sets.

The result follows from Cardinality of Set of All Mappings.

Also see

 * Count of Truth Functions
 * Binary Truth Functions