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\noindent
\textbf{Name:} \smallskip  \\
\textbf{Collaborators:} \smallskip \\ %%% List anyone with whom you discussed the problems
\textbf{Outside resources:} \smallskip \\ %%% List all resources used OTHER than the textbook, lecture notes, quizzes, worksheets, and previous homework assignments.
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\begin{center}
{
Math 2106, Foundations of Mathematical Proof \\
HW 5 --- Due March 15, 2017 (Wednesday)
}
\end{center}


\noindent From the textbook: \\
12.1, problems 6, 8, 10, 12.\\
12.2, problems 6, 10.\\
12.4, problems 8, 10.\\
12.5, problems 4, 8.\\

\noindent{\bf Additional problems}
\begin{enumerate}[label=A\arabic*]
\item Prove that the function $f : \NN \times \NN \rightarrow \NN$ given by $f(m,n) = 2^{m-1}(2n-1)$ is bijective. \\ (Let $\NN = \{1,2,3,\dots\}$.)
\item Determine whether each of the following relations from $\ZZ_6$ to $\ZZ_9$ is a function.  Justify your answers.
\begin{enumerate}
\item $\{([x]_6, [x]_9) \in \ZZ_6 \times \ZZ_9: x \in \ZZ\}$
\item $\{([x]_6, [2x]_9) \in \ZZ_6 \times \ZZ_9: x \in \ZZ\}$
\item $\{([x]_6, [3x]_9) \in \ZZ_6 \times \ZZ_9: x \in \ZZ\}$
\end{enumerate}
\item Let $f$ be a function from $A$ and $B$.  Let $C \subseteq A$ and $D \subseteq B$.  
\begin{enumerate}
\item Prove or disprove: $f^{-1}(f(C)) \subseteq C$.
\item Prove or disprove: $f^{-1}(f(C)) \supseteq C$.
\item Prove or disprove: $f(f^{-1}(D)) \subseteq D$.
\item Prove or disprove: $f(f^{-1}(D)) \supseteq D$.
\item Prove that $f$ is injective if and only if $f^{-1}(f(C)) = C$ for all $C \subseteq A$.
\item Prove that $f$ is surjective if and only if $f(f^{-1}(D))= D$ for all $D \subseteq B$.
\end{enumerate}
\end{enumerate}


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