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\lhead{\large \textbf{Math 461}}
\chead{\Large \textbf{Homework 2}}
\rhead{\large \textbf{DUE 9/15/2023}}
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{\Large\underline{\textbf{Exercises:}}} (\S $6$ and \S $7$)



\begin{enumerate}

\item Let $A$ and $B$ be non-empty sets.
	\begin{enumerate}
	\item Show that if $B$ is not finite and $B \subset A$, then $A$ is not finite.
	
	\item Prove that $A\times B$ is finite if and only if $A$ and $B$ are both finite.
	
	\item Show that $A \times B$ and $B \times A$ have the same cardinality.
	
	\item Let $B^A$ denote the set of functions $f\colon A\to B$. Show that if $A$ and $B$ are finite, then so is $B^A$.
	
	\item Suppose $B^A$ is finite and $B$ has at least two elements. Show that $A$ and $B$ are finite.
	\end{enumerate}

		
\item Let $\{0,1\}^\N$ denote the set of functions $f\colon \N\to \{0,1\}$.
	\begin{enumerate}
	\item Show that $\{0,1\}^\N$ and $\mathcal{P}(\N)$ have the same cardinality.
	
	
	\item Let $\mathcal{C}$ be the collection of \emph{countable} subsets of $\{0,1\}^\N$. Show that $\mathcal{C}$ and $\{0,1\}^\N$ have the same cardinality. [\textbf{Hint:} first construct an injection from $\mathcal{C}$ to $(\{0,1\}^\N)^\N$, and then use Exercise 5 from Homework 1.]
	
	\end{enumerate}

\item Let $n \in \N$.  In this problem we will prove that $\R^n$ has the same cardinality as $(0,1) \subset \R$.
	\begin{enumerate}
	
	\item Show that $\R$ has the same cardinality as $(0,1)$.
	
	\item Show that $(0,1) \times (0,1)$ has the same cardinality as $(0,1)$.  Conclude that $\R^2$ has the same cardinality as $\R$ as a result.
	
	\item Use, e.g., induction to prove that $\R^n$ has the same cardinality as $\R$.  Conclude that $\R^n$ has the same cardinality as $(0,1)$ as a result.
	
	\end{enumerate}

	
\item Let $\epsilon > 0$.  Prove that $(-\epsilon,\epsilon) \times \N  \times \{ 0,1 \}$ has the same cardinality as $\R^{10^{1000}}$.	
	
	
\end{enumerate}




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