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\noindent
\parbox{2.25in}{\bf Math 4710/5710-101}
\hfill {\Large \bf Homework \#3} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Sept. $10^\mathrm{th}$, 2014}

\vspace{0.4in}

\noindent {\sc\large Cardnality:} If there is an injection from $X$ into $\mathbb{N}=\mathbb{Z}_{\geq 0}$ (the natural numbers), we say that $X$ is {\bf countable}. If there is a bijection between $X$ and $\mathbb{N}$, then $X$ is countably {\it infinite} otherwise it is {\it finite}. 

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\noindent Munkres' \S7 problem \#5 (page 51). Determine which sets are countable. Justify your answers!
\begin{enumerate}[(a)]
   \item The set $A$ of all functions $f:\{0,1 \} \to \mathbb{Z}_{>0}$.
   \item {}[Let $n$ be some fixed positive integer.] The set $B_n$ of all functions $f:\{1,\dots,n\} \to \mathbb{Z}_{>0}$.
   \item The set $\displaystyle C = \bigcup\limits_{n \in \mathbb{Z}_{>0}} B_n$.
   \item The set $D$ of all functions $f:\mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$.
   \item The set $E$ of all functions $f:\mathbb{Z}_{>0} \to \{0,1\}$.
   \item The set $F$ of all functions $f:\mathbb{Z}_{>0} \to \{0,1\}$ that are {\bf eventually zero}. This means that there is some $N>0$ such that $f(x)=0$ for all $x>N$.
   \item The set $G$ of all functions $f:\mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ that are eventually 1.
   \item The set $H$ of all functions $f:\mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ that are eventually constant.
   \item The set $I$ of all two-element subsets of $\mathbb{Z}_{>0}$.
   \item The set $J$ of all finite subsets of $\mathbb{Z}_{>0}$.   
\end{enumerate}

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Grad students: Do all of these. Undergrads: You may skip 2 of your choice.

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