Definition:Diagonal Mapping

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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.