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\documentclass{math215}

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\doclabel{Math 215: Homework 14}
\docauthor{Your name here}
\docdate{May 3, 2013}

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\begin{document}

If $A$ and $B$ are sets, we say they have the {\it same cardinality}
if there exists a bijection $f:A\rightarrow B$, in which case we write
$|A|=|B|$.  We say a 
set $A$ is {\it countable} if there exists a bijection $f:\Nats\rightarrow A$.

\begin{proposition}{HW14.1} Cardinality defines an equivalence relation.
\end{proposition}
\begin{proof}
\end{proof}

\begin{proposition}{HW14.2} The set $(0,1)$ has the same cardinality
as $(-1,1)$.
\end{proposition}
\begin{proof}
\end{proof}

\begin{proposition}{HW14.3} The set $(0,1)$ has the same cardinality
as $(0,\infty)$.
\end{proposition}
\begin{proof}
\end{proof}

\begin{proposition}{HW14.4} The set $(-1,1)$ has the same cardinality
as $\Reals$.
\end{proposition}
\begin{proof}
\end{proof}

\begin{proposition}{HW14.5} The set $(0,1)$ has the same 
cardinality as $(0,1]$.
\end{proposition}
\begin{proof}
\end{proof}

\begin{proposition}{HW14.6} $\{a+\sqrt{2}b:a,b\in\Rats\}$ is countable.
\end{proposition}
\begin{proof}
\end{proof}

\end{document}