\begin{definition}[Definition:Diagonal Relation]
Let $S$ be a set.
The '''diagonal relation on $S$''' is a relation $\Delta_S$ on $S$ such that:
:$\Delta_S = \set {\tuple {x, x}: x \in S} \subseteq S \times S$
Alternatively:
:$\Delta_S = \set {\tuple {x, y}: x, y \in S: x = y}$
\end{definition}