handouts/Advanced/Size of Sets/parts/4 enumeration.tex

\section{Enumerations}

\definition{}
Let $A$ be a set. An \textit{enumeration} is a bijection from $A$ to $\{1, 2, ..., n\}$ or $\mathbb{N}$.\par
An enumeration assigns an element of $\mathbb{N}$ to each element of $A$.

\definition{}
We say a set is \textit{countable} if it has an enumeration.\par
We consider the empty set trivially countable.


\problem{}
Find an enumeration of $\{\texttt{A}, \texttt{B}, ..., \texttt{Z}\}$.
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\problem{}
Find an enumeration of $\mathbb{N}$.
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\problem{}
Find an enumeration of the set of squares $\{1, 4, 9, 16, ...\}$.

\problem{}
Let $A$ and $B$ be equinumerous sets. \par
Show that $A$ is countable iff $B$ is countable.

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\problem{}
Show that $\mathbb{Z}$ is countable.
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\problem{}<naturaltwo>
Show that $\mathbb{N}^2$ is countable.
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\problem{}
Show that $\mathbb{Q}$ is countable.
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\problem{}<naturalk>
Show that $\mathbb{N}^k$ is countable.
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\problem{}
Show that if $A$ and $B$ are countable, $A \cup B$ is also countable.\par
\note{Note that this automatically solves \ref{naturaltwo} and \ref{naturalk}.}
\vfill

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