injective functionInjectiveFunction2001-10-20 22:52:492007-02-19 13:25:29Definition\usepackage{amssymb}
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\usepackage{xypic}We say that a function $f\colon A\to B$ is \emph{injective} or \emph{one-to-one} if $f(x)=f(y)$ implies $x=y$, or equivalently, whenever $x\neq y$, then $f(x)\neq f(y)$.
\subsubsection*{Properties}
\begin{enumerate}
\item Suppose $A,B,C$ are sets and $f\colon A\to B$, $g\colon B\to C$
are injective functions. Then the composition $g\circ f$ is an injection.
\item Suppose $f\colon A\to B$ is an injection, and $C\subseteq A$. Then
the restriction $f|_C\colon C\to B$ is an injection.
\end{enumerate}
For a list of other \PMlinkescapetext{properties} of
injective functions, see \cite{wiki}.
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\bibitem{wiki} Wikipedia, article on \PMlinkexternal{Injective function}{http://en.wikipedia.org/wiki/Injective_function}.
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