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\title[Math 250 Handout 9 - functions and images]{Math 250 Handout 9  - functions and images}\maketitle

\begin{enumerate}
\item Give an example of a function with each of the following
  domain/codomains.
  \begin{enumerate}
  \item $\mathbf{Z} \to \mathbf{Z} \times \mathbf{Z}$.
  \item $\mathbf{Z} \times \mathbf{Z} \to \mathbf{Z}$.
  \item $\mathbf{Z} \to \mathbf{Z}_{>0}$.
  \item $\mathbf{R} \to \mathbf{Z}$.
  \item $P(\mathbf{R}) \to P(\mathbf{Z})$.
  \item $\mathbf{Z} \to P(\mathbf{Z})$.
  \item $P(\mathbf{Z}) \to \mathbf{Z}$.
  \end{enumerate}
  \vspace{11cm}


\item Draw a picture of two different functions from $\{1,2,3\} \to \{4,5\}$.


\newpage 

\begin{center}
  \textbf{Images}
\end{center}




\item Let $f\colon A \to B$ be a function. Finish the following
  sentence: an element $b \in B$ is not in the image of $f$ if
  $\cdots$\\\vspace{2cm}


\item Compute the image of the following functions:
  \begin{enumerate}

  \item $g \colon \mathbf{Z} \to \mathbf{Z}$, where $g(x) = 2n + 1$;
  \item $g \colon \mathbf{R} \to \mathbf{R}$, where $g(x) = 2x + 3$;
  \item $f \colon \mathbf{R} \to \mathbf{R}$, where $f(x) = -x^2 + 1$;
  \item $\cos \colon \mathbf{R} \to \mathbf{R}$;
  \item $\tan^{-1} \colon \mathbf{R} \to \mathbf{R}$;
  \end{enumerate}
\vspace{1in}






\item   Let $f\colon A \to B$ be a function and let $X,Y \subset A$. Prove
  or disprove each of the following:
  \begin{enumerate}
  \item $X \subset Y \Rightarrow f(X) \subset f(Y)$.
  \item $X \subset Y \Leftarrow f(X) \subset f(Y)$.
  \item $f(X \cup Y) \subset f(X) \cup f(Y)$.
  \item $f(X \cup Y) \supset f(X) \cup f(Y)$.
  \item $f(X \cap Y) \subset f(X) \cap f(Y)$.
  \item $f(X \cap Y) \supset f(X) \cap f(Y)$.
  \item (HW) $f(X) - f(Y) \subset f(X - Y)$.
  \item (HW) $f(X) - f(Y) \supset f(X - Y)$.
\end{enumerate}  

  \end{enumerate}
\end{document}




