Definition:Surjection/Definition 1

Definition
Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping from $S$ to $T$.

$f: S \to T$ is a surjection iff:
 * $\forall y \in T: \exists x \in \operatorname{Dom} \left({f}\right): f \left({x}\right) = y$

That is, iff $f$ is right-total.

Thus a surjection is a relation which is:
 * Left-total
 * Many-to-one
 * Right-total.

Also see

 * Equivalence of Definitions of Surjection