Cardinality of Set of Endorelations

Theorem

Let $S$ be a finite set.

Let $\card S = n$, where $\card {\, \cdot \,}$ denotes cardinality (that is, the number of elements).

Let $\RR$ denote the set of all endorelations on $S$.

Then the cardinality of $\RR$ is given by:

$\card \RR = 2^{\paren {n^2} }$

Proof

By definition, an endorelation on $S$ is a relation from $S$ to itself.

From Cardinality of Set of Relations, the number of relations from $S$ to $T$ is equal to $2^{\card S \times \card T}$.

In this case $S = T$ and the result follows.

$\blacksquare$