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\noindent
\parbox{1.95in}{\bf Math 4710/5710}
\hfill {\Large \bf Homework \#3} \hfill
\parbox{1.95in}{\bf \hfill Due: Mon., Feb. 3${}^\mathbf{rd}$\!, 2025}

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%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force

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Recall that if there is an injection from a set $A$ to a set $B$, we say $|A| \leq |B|$ (i.e., $A$'s cardinality is no more than $B$'s cardinality). If there is a bijection between $A$ and $B$, we say $|A|=|B|$ (i.e., $A$ and $B$ have the same cardinality). If there is an injection from a set $A$ into $\mathbb{N}=\mathbb{Z}_{\geq 0}$ (i.e., the natural numbers = non-negative integers), then we say $A$ is {\bf countable}. If there is a bijection between $A$ and $\mathbb{N}$, we say $A$ is {\bf countably infinite}. If $A$ is countable but not countably infinite, we say $A$ is {\bf finite}.

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{\it Note:} We have shorthand notations for countably infinite and continuum (i.e., the size of the real numbers) cardinalities. In particular, $|\mathbb{N}| = \aleph_0$ and $|\mathbb{R}|=2^{\aleph_0}=\mathfrak{c}$.

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\begin{description}
\item[\#1 A Finite Number of Countable Questions:] Determine which sets are countable. Explain your answers.

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Undergrads: You may skip 2 of your choice. {\bf Grad students:} Do them all.

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\begin{enumerate}[(a)]
   \item The set $A$ of all functions $f:\{0,1 \} \to \mathbb{N}$.

   \item The set $B_n$ of all functions $f:\{1,\dots,n\} \to \mathbb{N}$ where $n$ is some fixed positive integer. 
   
           {\it Note:} $B_0$ would be all functions from the empty set to $\mathbb{N}$. The only such function would be the empty function, so $|B_0|=1$. 
           In particlar, $B_0$ is countable but finite\dots{}and a weirdo. That's right -- you heard me -- a weirdo!
           
   \item The set $\displaystyle C = \bigcup\limits_{n \in \mathbb{N}}^\infty B_n$.
    
   \item The set $D$ of all functions $f:\mathbb{N} \to \mathbb{N}$.

   \item The set $E$ of all functions $f:\mathbb{N} \to \{0,1\}$.
   
   \item The set $F$ of all functions $f:\mathbb{N} \to \{0,1\}$ that are {\bf eventually zero} (i.e., for each $f$ there is some $N>0$ such that $f(x)=0$ for all $x>N$).
      
   \item The set $G$ of all functions $f:\mathbb{N} \to \mathbb{N}$ that are eventually 1.
   
   \item The set $H$ of all functions $f:\mathbb{N} \to \mathbb{N}$ that are eventually constant.
   
   \item The set $I$ of all two-element subsets of $\mathbb{N}$.
   
   \item The set $J$ of all finite subsets of $\mathbb{N}$.   
\end{enumerate}

\item[\#2 Infinite Calculations:] Calculate cardinalities.\\ \force\hfill Justify your answers using basic cardinal arithmetic and inequalities. 

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{\it Note:} Your answers should either be a non-negative integer (0 or 1 or 2 or \dots), $\aleph_0$, $\mathfrak{c}=2^{\aleph_0}$, or $2^\mathfrak{c}=2^{2^{\aleph_0}}$.

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For example: $2^{\aleph_0} \leq \aleph_0^{\aleph_0} \leq (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0^2}=2^{\aleph_0}$ so that $|\mathbb{Z}^\mathbb{Z}|=|\mathbb{Z}|^{|\mathbb{Z}|}=\aleph_0^{\aleph_0} = 2^{\aleph_0}=\mathfrak{c}$.

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\begin{enumerate}[(a)]
\item $|10^{\mathcal{P}(\mathbb{Z})}|$

\item $|\mathbb{Z}\times \mathbb{R}^3|+|\mathbb{Q}^\mathbb{N}|$

\item Consider $\mathbb{N}^5$ vs. $5^{\mathbb{N}}$. Do these sets have the same cardinality? Are they countable? (both, either, neither?)

\end{enumerate}
\end{description}

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