surjective

A function $f\colon X\to Y$ is called surjective or onto if, for every $y\in Y$ , there is an $x\in X$ such that $f(x)=y$ .

Equivalently, $f\colon X\to Y$ is onto when its image is all the codomain: $$\mathrm{Im} f= Y.$$

Properties

1. If $f\colon X\to Y$ is any function, then $f\colon X\to f(X)$ is a surjection. That is, by restricting the codomain, any function induces a surjection.
2. The composition of surjective functions (when defined) is again a surjective function.
3. If $f\colon X\to Y$ is a surjection and $B\subseteq Y$ , then (see this page) $$ f f^{-1}(B) = B. $$