diff --git a/Advanced/Size of Sets/main.tex b/Advanced/Size of Sets/main.tex index 557212a..0f8c25c 100755 --- a/Advanced/Size of Sets/main.tex +++ b/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. diff --git a/Advanced/Size of Sets/parts/0 sets.tex b/Advanced/Size of Sets/parts/0 sets.tex index d01df7a..308fc46 100644 --- a/Advanced/Size of Sets/parts/0 sets.tex +++ b/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 diff --git a/Advanced/Size of Sets/parts/3 functions.tex b/Advanced/Size of Sets/parts/3 functions.tex index 9b8bc18..f8c4fb6 100644 --- a/Advanced/Size of Sets/parts/3 functions.tex +++ b/Advanced/Size of Sets/parts/3 functions.tex @@ -372,4 +372,4 @@ Intuitively, you can think of a bijection as a \say{matching} between elements o \definition{} We say two sets $A$ and $B$ are \textit{equinumerous} if there exists a bijection $f: A \to B$. -\pagebreak +\pagebreak \ No newline at end of file diff --git a/Advanced/Size of Sets/parts/4 enumeration.tex b/Advanced/Size of Sets/parts/4 enumeration.tex index bf5d76f..f675126 100644 --- a/Advanced/Size of Sets/parts/4 enumeration.tex +++ b/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{} Show that $\mathbb{N}^2$ is countable. \vfill \problem{} +Show that $\mathbb{Q}$ 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. +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 \ No newline at end of file diff --git a/Advanced/Size of Sets/parts/dense.tex b/Advanced/Size of Sets/parts/5 dense.tex similarity index 97% rename from Advanced/Size of Sets/parts/dense.tex rename to Advanced/Size of Sets/parts/5 dense.tex index b3fbb87..44f33bc 100644 --- a/Advanced/Size of Sets/parts/dense.tex +++ b/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 diff --git a/Advanced/Size of Sets/parts/6 uncountable.tex b/Advanced/Size of Sets/parts/6 uncountable.tex new file mode 100644 index 0000000..17ff9ec --- /dev/null +++ b/Advanced/Size of Sets/parts/6 uncountable.tex @@ -0,0 +1,74 @@ +\section*{Uncountable Sets} + +\problem{} +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 \ No newline at end of file