Set edits

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

@@ -2,7 +2,7 @@
 
 \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 $A$.
+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
@@ -31,16 +31,21 @@ Show that $A$ is countable iff $B$ is countable.
 Show that $\mathbb{Z}$ is countable.
 \vfill
 
-\problem{}
+\problem{}<naturaltwo>
 Show that $\mathbb{N}^2$ is countable.
 \vfill
 
 \problem{}
+Show that $\mathbb{Q}$ is countable.
+\vfill
+
+\problem{}<naturalk>
 Show that $\mathbb{N}^k$ is countable.
 \vfill
 
 \problem{}
-Show that if $A$ and $B$ are countable, $A \cup B$ is also countable.
+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
 
 \pagebreak