Definition 1. A surjective function or surjection is a a function $f : X \to Y$ such that for all $y \in Y$,
\[\exists x \in X : f(x) = y.\]In other words, $\im f = Y$.
The set of all surjections from $X$ onto $Y$ is denoted by
\[\sur(X, Y).\]Note. Surjective is also known as onto.