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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 6 --- Due April 3, 2017 (Monday)
}
\end{center}

\begin{center}
\begin{tabular}{ll}
\begin{minipage}{0.3\textwidth}
From Hammack's book:\\
13.1: 14.\\  13.2: 4, 6, 12. \\ 13.3: 2, 4, 6, 8, 10. \\ 13.4: 2, 4.\\
\end{minipage}
&
\begin{minipage}{0.3\textwidth}
From Ross' book:\\
7.2, 7.4, \\ 8.2(e), 8.4, 8.6, 8.8(b), 8.9.
\end{minipage}
\end{tabular}
\end{center}


\noindent{\bf Additional problems}
\begin{enumerate}[label=A\arabic*]
\item For each of the following sets, determine whether it is countable.  Justify your answers.
\begin{enumerate}
\item The set of all subsets of $\NN$.
\item The set of finite subsets of $\NN$.
\item The set of all squences of natural numbers.
\item The set of sequences of natural numbers with finitely many non-zero entries.
\end{enumerate}
\item \begin{enumerate}
\item Prove that $|\RR^\RR| \leq |\cP(\RR \times \RR)|$.
\item Prove that $|\{0,1\}^\RR| \leq |\RR^\RR|$.
\item Prove that $|\RR^\RR| = |\cP(\RR)|$.
\item Prove that $|\NN^\RR| = |\cP(\RR)|$.
\end{enumerate}
Recall that $B^A$ denotes the set of all functions from $A$ to $B$.
\item Let $\II$ be the set of irrational numbers, i.e. $\II = \RR - \QQ$.
\begin{enumerate}
\item Prove that $\II$ is uncountable.
\item Prove that $|\II| = |\RR|$ without using the continuum hypothesis. That is, do not assume that any uncountable subset of $\RR$ has the same cardinality has $\RR$.
\end{enumerate}
\end{enumerate}


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