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

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+\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 assignes 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 $\{\texttt{A}, \texttt{B}, ..., \texttt{Z}\}$.
+\vfill
+
+\problem{}
+Find an enumeration of $\mathbb{N}$.
+\vfill
+
+\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.
+
+\vfill
+\pagebreak
+
+\problem{}
+Show that $\mathbb{Z}$ is countable.
+\vfill
+
+\problem{}
+Show that $\mathbb{N}^2$ is countable.
+\vfill
+
+\problem{}
+Show that $\mathbb{N}^k$ is countable.
+\vfill
+
+\problem{}
+Show that if $A$ and $B$ are countable, $A \cup B$ is also countable.
+\vfill
+
+\pagebreak