surjectiveSurjective2002-03-14 00:08:482008-05-25 12:10:33Definitionsurjection%\usepackage{graphicx}
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\newcommand{\mathbb}[1]{\mathbbmss{#1}}A function $f\colon X\to Y$ is called \emph{surjective} or \emph{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:
$$\mathrm{Im} f= Y.$$
\subsubsection*{Properties}
\begin{enumerate}
\item 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.
\item The composition of surjective functions (when defined) is
again a surjective function.
\item If $f\colon X\to Y$ is a surjection and $B\subseteq Y$, then
(see \PMlinkname{this page}{InverseImage})
$$
f f^{-1}(B) = B.
$$
\end{enumerate}