Set edits

Advanced/Size of Sets/main.tex

@@ -1,13 +1,13 @@
 % use [nosolutions] flag to hide solutions.
 % use [solutions] flag to show solutions.
 \documentclass[
-	solutions,
+	nosolutions,
 	singlenumbering
 ]{../../resources/ormc_handout}
 
 \uptitlel{Advanced 1}
 \uptitler{Summer 2023}
-\title{The Size of Sets, Part 1}
+\title{The Size of Sets}
 \subtitle{Prepared by Mark on \today{}}
 
 \begin{document}
@@ -19,7 +19,8 @@
 	\input{parts/2 cartesian.tex}
 	\input{parts/3 functions.tex}
 	\input{parts/4 enumeration.tex}
-	\input{parts/dense.tex}
+	%\input{parts/5 dense.tex}
+	\input{parts/6 uncountable.tex}
 
 
 	%\vfill
@@ -36,14 +37,6 @@
 	%	$(a, b) = \{ \{a\}, \{a, b\}\}$
 	%\end{solution}
 
-	%\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{}
 	%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
 	%Provide a proof or a counterexample.

Advanced/Size of Sets/parts/0 sets.tex

@@ -22,11 +22,13 @@ Note that the \say{subset} symbol resembles the \say{less than or equal to} symb
 
 We can also write $\{a, b\} \subset \{a, b, c\}$, which denotes a \textit{strict subset.} \par
 The relationship between $\subseteq$ and $\subset$ is the same as the relationship between $\leq$ and $<$. \par
+In particular, if $A \subset B$, $A \subseteq B$ and $A \neq B$ \par
 For example, $\{a, b, c\} \subseteq \{a, b, c\}$ is true, but $\{a, b, c\} \subset \{a, b, c\}$ is false.
 
 
 \definition{}
 The \textit{empty set}, usually written $\varnothing$, is the unique set containing no elements. \par
+By definition, the empty set is a subset of every set. \par
 \note[Note]{The $\varnothing$ symbol is called \say{varnothing.} If you'd like to know why, ask an instructor.}
 
 \problem{}
@@ -70,6 +72,11 @@ What is the power set of $\{1, 2, 3\}$? \par
 Let $A$ be a set with $n$ elements. \par
 How many elements does $\mathcal{P}(A)$ have? \par
 \hint{Binary may help.}
+\vfill
+
+\problem{}
+Show that the set of all sets that do not contain themselves is not a set. \par
+
 
 \vfill
 \pagebreak

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

Advanced/Size of Sets/parts/5 dense.tex

@@ -1,4 +1,4 @@
-\section*{Bonus: Dense Orderings}
+\section{Dense Orderings}
 
 \definition{}
 An \textit{ordered set} is a set with an \say{order} attached to it. \par

Advanced/Size of Sets/parts/6 uncountable.tex

@@ -0,0 +1,74 @@
+\section*{Uncountable Sets}
+
+\problem{}<binarystrings>
+Let $B$ be the set of infinite binary strings. Show that $B$ is not countable. \par
+Here's how you should start:
+
+\vspace{2mm}
+
+Assume we have some enumeration $n(b)$ that assigns a natural number to every $b \in B$.\par
+Now, arrange the elements of $B$ in a table, in order of increasing index: \par
+
+\begin{center}
+\begin{tikzpicture}[scale=0.5]
+	\node at (0, 0) {$n(b)$};
+	\node at (4.5, 0) {digits of $b$};
+
+	% Vertical lines
+	\draw (1, 0.5) -- (1, -8);
+	\draw (-1, 0.5) -- (-1, -8);
+
+	% Horizontal title
+	\draw (-1, -0.5) -- (8, -0.5);
+
+	\foreach \i/\j in {
+		0/1010100110011110,
+		1/0101101011010010,
+		2/1101011001010101,
+		3/0001100101010110,
+		4/1101011101000110,
+		5/1101100010100111,
+		6/1011001101001010%
+	} {
+		\node at (0, -\i-1) {$\i$};
+		\draw (-1, -1.5 - \i) -- (8, -1.5 - \i);
+		\node[anchor=west] at (1, -\i-1) {\texttt{\j}...};
+	}
+
+	\node at (0, -7-1) {...};
+	\node at (4.5, -7-1) {.....};
+\end{tikzpicture}
+\end{center}
+
+
+First, convince yourself that if $B$ is countable, this table will contain every element of $B$, \par
+then construct a new element of $B$ that is guaranteed to \textit{not} be in this table.\par
+\hint{What should the first digit of this new string be? What should its second digit be? \\
+Or, even better, what \textit{shouldn't} they be?}
+\vfill
+
+
+\problem{}
+Using \ref{binarystrings}, show that $\mathcal{P}(\mathbb{N})$ is uncountable.
+\vfill
+\pagebreak
+
+\problem{}
+Show that $\mathbb{R}$ is not countable. \par
+\hint{Earlier in this handout, we defined a real number as \say{a decimal, finite or infinite.}}
+\vfill
+
+\problem{}
+Find a bijection from $(0, 1)$ to $\mathbb{R}$.\par
+\hint{$(0, 1)$ is the set of all real numbers between 0 and 1, not including either endpoint.}
+\vspace{2mm}
+This problem brings us to the surprising conclusion that there are \say{just as many} numbers between 0 and 1 as there are in the entire real line.
+
+\vfill
+
+\problem{}
+Find a bijection between $(0, 1)$ and $[0, 1]$. \par
+\hint{$[0, 1]$ is the set of all real numbers between 0 and 1, including both endpoints.}
+\vfill
+
+\pagebreak