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\begin{document}

\title[Math 250 Handout 11 - injectivity and surjectivity warmup]{Math 250 Handout 11 - injectivity}\maketitle





\begin{enumerate}

\item Draw a picture of a function which is 
  \begin{enumerate}
  \item injective\vspace{3cm}
  \item not injective\vspace{3cm}
  \end{enumerate}

\item Finish the following sentence: a function $f\colon X \to Y$ is
  \emph{not} injective if $\ldots$\\ \vspace{3cm}





\newpage
\item Which of the following functions are injective?\vspace{22pt}


  \begin{enumerate}
  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{R} \to \mathbf{R};\, x \mapsto x^2$.
\vspace{22pt}

  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{R} \to \mathbf{R};\, x \mapsto \frac{x+1}{2}$.
\vspace{22pt}

  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{R} \to \mathbf{R};\, x \mapsto \cos{x}$.
\vspace{22pt}

  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$[0,\pi) \to \mathbf{R};\, x \mapsto \cos{x}$.
\vspace{22pt}


  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{R}_{\geq 0} \to \mathbf{R};\, x \mapsto \frac{x-1}{x+1}$.
\vspace{22pt}


  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{R}-\{-1\} \to \mathbf{R};\, x \mapsto \frac{x-1}{x+1}$.
\vspace{22pt}




  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{R} \to \mathbf{R};\, x \mapsto \arctan{x}$.
\vspace{22pt}


  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{R}^3 \to \mathbf{R}^2;\, (x,y,z)\mapsto (x,y)$.
\vspace{22pt}

  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{R}^2 \to \mathbf{R}^3;\, (x,y)\mapsto (x+y,x-y,x^2 + y^2)$.
\vspace{22pt}




  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$P(\mathbf{R}) \to P(\mathbf{Z});\, S \mapsto S \cap \mathbf{Z}$.
\vspace{22pt}

  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$P(\mathbf{Z}) \to P(\mathbf{Z});\, S \mapsto S \cup \{1\}$.
\vspace{22pt}


  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{Z} \to P(\mathbf{Z});\, n \mapsto \{n\}$.
\vspace{22pt}

  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$P(\mathbf{Z}) \to \mathbf{Z};\, S \mapsto |S| \text{ if } S \text{ is finite}, 0
\text{ if } S \text{ is infinite}$.
\vspace{22pt}

  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{R} \to \mathbf{R};\, x \mapsto x^3+1$.
\vspace{22pt}

  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{R} \to \mathbf{R};\, x \mapsto x(x^2-1)$.
\vspace{22pt}

  \item \textbf{I}nj\,  \textbf{N}:
    \hspace{12pt}
$\mathbf{C} \to \mathbf{C};\, x \mapsto x^2$.
\vspace{22pt}

  \item \textbf{I}nj\, \textbf{N}:
    \hspace{12pt}
$[1,\infty) \to [0,\infty);\, x \mapsto x^3 - x$.
\vspace{22pt}

  \end{enumerate}

\end{enumerate}






\end{document}




