Edits

Advanced/Size of Sets/main.tex

@@ -18,44 +18,43 @@
 	\input{parts/1 really big.tex}
 	\input{parts/2 cartesian.tex}
 	\input{parts/3 functions.tex}
-	\input{parts/4 dense.tex}
+	\input{parts/4 enumeration.tex}
+	\input{parts/dense.tex}
 
 
-	\vfill
+	%\vfill
-	\pagebreak
+	%\pagebreak
 
-	\section{Bonus Problems}
+	%\section{Bonus Problems}
 
-	\problem{}
+	%\problem{}
-	Using only sets, how can we build an ordered pair $(a, b)$? \par
+	%Using only sets, how can we build an ordered pair $(a, b)$? \par
-	$(a, b)$ should be equal to $(c, d)$ if and only if $a = b$ and $c = d$. \par
+	%$(a, b)$ should be equal to $(c, d)$ if and only if $a = b$ and $c = d$. \par
-	Of course, $(a, b) \neq (b, a)$.
+	%Of course, $(a, b) \neq (b, a)$.
 
-	\begin{solution}
+	%\begin{solution}
-		$(a, b) = \{ \{a\}, \{a, b\}\}$
+	%	$(a, b) = \{ \{a\}, \{a, b\}\}$
-	\end{solution}
+	%\end{solution}
 
-	\vfill
+	%\vfill
 
+	%\problem{}
+	%Let $R$ be the set of all sets that do not contain themselves. \par
+	%Does $R$ exist? \par
+	%\hint{If $R$ exists, do we get a contradiction?}
+	%\vfill
 
-	\problem{}
+	%\problem{}
-	Let $R$ be the set of all sets that do not contain themselves. \par
+	%Suppose $f: A \to B$ and $g: B \to C$ are both one-to-one. Must $h(x) = g(f(x))$ be one-to-one? \par
-	Does $R$ exist? \par
+	%Provide a proof or a counterexample.
-	\hint{If $R$ exists, do we get a contradiction?}
-	\vfill
 
+	%\vfill
 
-	\problem{}
+	%\problem{}
-	Suppose $f: A \to B$ and $g: B \to C$ are both one-to-one. Must $h(x) = g(f(x))$ be one-to-one? \par
+	%Suppose $f: A \to B$ and $g: B \to C$ are both onto. Must $h(x) = g(f(x))$ be onto? \par
-	Provide a proof or a counterexample.
+	%Provide a proof or a counterexample.
 
-	\vfill
+	%\vfill
-
+	%\pagebreak
-	\problem{}
-	Suppose $f: A \to B$ and $g: B \to C$ are both onto. Must $h(x) = g(f(x))$ be onto? \par
-	Provide a proof or a counterexample.
-
-	\vfill
-	\pagebreak
 
 \end{document}

Advanced/Size of Sets/parts/3 functions.tex

@@ -205,12 +205,6 @@ Is this function one-to-one? Is it onto?
 
 \vfill
 
-\problem{}
-Consider the function $f: \mathbb{N} \to \mathbb{N}$ defined by $f(x) = x^2$. \par
-Is this function one-to-one? Is it onto?
-
-\vfill
-
 \problem{}
 Consider the function $f: \mathbb{Z} \to \mathbb{Z}$ defined below. \par
 Is this function one-to-one? Is it onto?
@@ -375,4 +369,7 @@ Show that no bijection between $A$ and $B$ exists.
 
 Intuitively, you can think of a bijection as a \say{matching} between elements of $A$ and $B$. If we were to draw a bijection, we'd see an arrow connecting every element in $A$ to every element in $B$. If a bijection exists, every element of $A$ directly corresponds to an element of $B$, therefore $A$ and $B$ must have the same number of elements.
 
+\definition{}
+We say two sets $A$ and $B$ are \textit{equinumerous} if there exists a bijection $f: A \to B$.
+
 \pagebreak

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

@@ -0,0 +1,46 @@
+\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

Advanced/Size of Sets/parts/dense.tex

@@ -1,6 +1,4 @@
-\section{Dense Orderings}
+\section*{Bonus: Dense Orderings}
-
-\note[Note]{This section is fairly difficult. If you get stuck here, try the bonus problems on the next page.}
 
 \definition{}
 An \textit{ordered set} is a set with an \say{order} attached to it. \par
@@ -17,11 +15,13 @@ We say an ordered set $A$ is \textit{dense} if for any $a, b \in A$ there is a $
 Intuitively, this means that there is an element of $A$ between any two elements of $A$.
 
 \problem{}
-Show that the ordered set $(\mathbb{Q}, <)$ is dense.
+Show that the ordered set $(\mathbb{Q}, <)$ is dense.\par
+\hint{Elements of $\mathbb{Q}$ are defined as fractions $\frac{p}{q}$, where $p$ and $q$ are integers.}
 \vfill
 
 \problem{}
-Show that the ordered set $(\mathbb{R}, <)$ is dense.
+Show that the ordered set $(\mathbb{R}, <)$ is dense.\par
+\hint{We can define a \say{real number} as a decimal, finite or infinite.}
 \vfill
 
 \problem{}