(related to Proposition: Number of Relations on a Finite Set)
According to the definition of relation on a set \(V\) a relation \(R\) is any subset of the Cartesian product \(R\subseteq V\times V\).
According to the fundamental counting principle the cardinality of the Cartesian product is \(|V\times V|=n^2\), if \(V\) is finite a finite set with \(|V|=n\). In particular, the set \(V\times V\) is finite.
It follows from the number of subsets of a given finite set that
\[|R\subseteq V\times V|=2^{n^2}.\]