\begin{definition}[Definition:Diagonal Mapping]
Let $S$ be a set.
Let $S \times S$ be the Cartesian product of $S$ with itself.
Then the '''diagonal mapping on $S$''' is defined as $\Delta: S \to S \times S$:
:$\forall x \in S: \Delta \left({x}\right) = \left({x, x}\right)$
Clearly $\Delta$ is an injection, and is not a surjection unless $S$ is a singleton.
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\end{definition}