From 2eef68075228a74721a7c45d61963e0d6742e324 Mon Sep 17 00:00:00 2001 From: Mark Date: Tue, 18 Jul 2023 10:11:04 -0700 Subject: [PATCH] Added set size draft --- Advanced/Size of Sets/main.tex | 61 +++ Advanced/Size of Sets/parts/0 sets.tex | 106 ++++++ Advanced/Size of Sets/parts/1 really big.tex | 65 ++++ Advanced/Size of Sets/parts/2 cartesian.tex | 172 +++++++++ Advanced/Size of Sets/parts/3 functions.tex | 378 +++++++++++++++++++ Advanced/Size of Sets/parts/4 dense.tex | 35 ++ 6 files changed, 817 insertions(+) create mode 100755 Advanced/Size of Sets/main.tex create mode 100644 Advanced/Size of Sets/parts/0 sets.tex create mode 100644 Advanced/Size of Sets/parts/1 really big.tex create mode 100644 Advanced/Size of Sets/parts/2 cartesian.tex create mode 100644 Advanced/Size of Sets/parts/3 functions.tex create mode 100644 Advanced/Size of Sets/parts/4 dense.tex diff --git a/Advanced/Size of Sets/main.tex b/Advanced/Size of Sets/main.tex new file mode 100755 index 0000000..f9208aa --- /dev/null +++ b/Advanced/Size of Sets/main.tex @@ -0,0 +1,61 @@ +% use [nosolutions] flag to hide solutions. +% use [solutions] flag to show solutions. +\documentclass[ + solutions, + singlenumbering +]{../../resources/ormc_handout} + +\uptitlel{Advanced 1} +\uptitler{Summer 2023} +\title{The Size of Sets, Part 1} +\subtitle{Prepared by Mark on \today{}} + +\begin{document} + + \maketitle + + \input{parts/0 sets.tex} + \input{parts/1 really big.tex} + \input{parts/2 cartesian.tex} + \input{parts/3 functions.tex} + \input{parts/4 dense.tex} + + + \vfill + \pagebreak + + \section{Bonus Problems} + + \problem{} + 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 + Of course, $(a, b) \neq (b, a)$. + + \begin{solution} + $(a, b) = \{ \{a\}, \{a, b\}\}$ + \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. + + \vfill + + \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} \ No newline at end of file diff --git a/Advanced/Size of Sets/parts/0 sets.tex b/Advanced/Size of Sets/parts/0 sets.tex new file mode 100644 index 0000000..d01df7a --- /dev/null +++ b/Advanced/Size of Sets/parts/0 sets.tex @@ -0,0 +1,106 @@ +\section{Set Basics} + +\definition{} +A \textit{set} is a collection of objects. \par +If $a$ is an element of set $S$, we write $a \in S$. This is pronounced \say{$a$ in $S$.} \par +The position of each element in a set or the number of times it is repeated doesn't matter. \par +All that matters is \textit{which} elements are in the set. + +\vspace{2mm} + +We say two sets $A$ and $B$ are equal if every element of $A$ is in $B$, and every element of $B$ is in $A$. This is known as the \textit{principle of extensionality.} + +\problem{} +Convince yourself that $\{a, b\} = \{b, a\} = \{a, b, a, b, b\}$. + +\definition{} +A set $A$ is a \textit{subset} of a set $B$ if every element of $A$ is in $B$. \par +For example, $\{a, b\}$ is a subset of $\{a, b, c\}$. This is written $\{a, b\} \subseteq \{a, b, c\}$. \par +Note that the \say{subset} symbol resembles the \say{less than or equal to} symbol. + +\vspace{2mm} + +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 +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 +\note[Note]{The $\varnothing$ symbol is called \say{varnothing.} If you'd like to know why, ask an instructor.} + +\problem{} +Which of the following are true? +\begin{itemize} + \item $\{1, 3\} = \{3, 3, 1\}$ + \item $\{1, 2\} \subset \{2\}$ + \item $\{1, 2\} \subset \{1, 2\}$ + \item $\{1, 2\} \subseteq \{1, 2\}$ + \item $\{2\} \subseteq \{1, 2\}$ + \item $\varnothing \subseteq \{1, 2\}$ +\end{itemize} + +\vfill +\pagebreak + +\problem{} +Let $A$ and $B$ be sets. Convince yourself that $A \subseteq B$ and $B \subseteq A$ implies $A = B$. + +\vspace{2mm} + +\hint{Whenever you start a proof, you should first look at definitions. \\ +As stated on the previous page, $A = B$ if every element in $A$ is in $B$ and every element of $B$ is in $A$.} + +\vspace{2mm} + +As we saw before, the $\subseteq$ relation behaves a lot like the $\leq$ relation. \par +The statement above is very similar to the statement \say{$x \leq y$ and $y \geq x$ implies $x = y$}. + + +\definition{} +Let $A$ be a set. The \textit{power set} of $A$, written $\mathcal{P}(A)$, is the set of all subsets of $A$. + +\problem{} +What is the power set of $\{1, 2, 3\}$? \par +\hint{It has eight elements.} + +\vfill + +\problem{} +Let $A$ be a set with $n$ elements. \par +How many elements does $\mathcal{P}(A)$ have? \par +\hint{Binary may help.} + +\vfill +\pagebreak + + +\definition{Set Operations} +$A \cap B$ is the \textit{intersection} of $A$ and $B$. \par +It is the set of objects that are in both $A$ and $B$. + +\vspace{3mm} + +$A \cup B$ is the \textit{union} of $A$ and $B$. \par +It is the set of objects that are in either $A$ or $B$. + +\vspace{3mm} + +$A - B$ is the \textit{difference} of $A$ and $B$. \par +It is the set of objects that are in $A$ but are not in $B$. + +\problem{} +What is $\{a, b, c\} \cap \{b, c, d\}$? + +\vfill + +\problem{} +What is $\{a, b, c\} \cup \{b, c, d\}$? + +\vfill + +\problem{} +What is $\{a, b, c\} - \{b, c, d\}$? + +\vfill +\pagebreak \ No newline at end of file diff --git a/Advanced/Size of Sets/parts/1 really big.tex b/Advanced/Size of Sets/parts/1 really big.tex new file mode 100644 index 0000000..ddc74b9 --- /dev/null +++ b/Advanced/Size of Sets/parts/1 really big.tex @@ -0,0 +1,65 @@ +\section{Really Big Sets} + +\definition{} +%We say a set $S$ is \textit{finite} if there exists a bijection from $S$ to $\{1, 2, 3, ..., n\}$ for some integer $n$. \par +%In other words +We say set is \textit{finite} if its elements can be consecutively numbered from 1 to some maximum index $n$. \par +Informally, we could say that a set is finite if it \say{ends.} \par +For example, the set $\{\star, \diamond, \heartsuit\}$ is (obviously) finite. We can number its elements 1, 2, and 3. + +\vspace{2mm} + +If a set is not finite, we say it is \textit{infinite}. + +\vspace{2mm} + + + +\problem{} +Which of the following sets are finite? +\begin{itemize} + \item $\{\texttt{A}, \texttt{B}, ..., \texttt{Z}\}$ + \item $\{ \text{all rats in Europe} \}$ + \item $\{ \text{all positive numbers} \}$ + \item $\{ \mathbb{ \text{all rational numbers} } \}$ +\end{itemize} + + +\vfill + +\generic{Remark:} +Note that our definition of \say{infinite-ness} is based on a property of the set. Saying \say{a set is infinite} is much like saying \say{a cat is black} or \say{a number is even}. There are many different kinds of black cats, and there are many different even numbers --- some large, some small. \par + +\vspace{2mm} + +In general, \textbf{$\infty$ is not a well-defined mathematical object\footnotemark{}}. Infinity is not a number. There isn't a single \say{infinity.} Infinity is the the general concept of endlessness, used in many different contexts. + +\vspace{2mm} + +%The Russian language (as well as many others, no doubt) captures this well: \say{infinity} in Russian is \say{бес-конеч-ность}, which can be literally translated as \say{without-end-ness}. + + +\footnotetext{ + In most cases. There are exceptions, but you need not worry about them for now. If you're curious, you may ask an instructor to explain. There's also a chance we'll see a well-defined \say{infinity} in a handout later this quarter. +} + + +\vfill +\pagebreak + +%Say we have two finite sets $A$ and $B$. Comparing the sizes of these is fairly easy: all we need to do is count the elements %in each. It is not difficult to see that $\{1, 2, 3\}$ is bigger than $\{1, 2\}$. +% +%\vspace{2mm} +% +%We could extend this notion of \say{size} to infinite sets. \par +%For example, consider $\mathbb{R}$ and $\mathbb{Z}$. Intuitively, we'd expect $\mathbb{R}$ to be larger, \par +%since there are many elements in $\mathbb{R}$ between every two elements in $\mathbb{Z}$. +% +%\vspace{1mm} +% +%We could also try to compare the sizes of $\mathbb{Q}$ and $\mathbb{Z}$. There are bIntuitively, we'd expect $\mathbb{R}$ to %be larger, \par +%since there are many elements in $\mathbb{R}$ between every two elements in $\mathbb{Z}$. +% +% +%\vfill +%\pagebreak \ No newline at end of file diff --git a/Advanced/Size of Sets/parts/2 cartesian.tex b/Advanced/Size of Sets/parts/2 cartesian.tex new file mode 100644 index 0000000..422ed03 --- /dev/null +++ b/Advanced/Size of Sets/parts/2 cartesian.tex @@ -0,0 +1,172 @@ +\section{Common Sets and Cartesian Products} + +\definition{} +There are a few sets we use often. They have special names: +\begin{itemize} + \item $\mathbb{N} = \{0, 1, 2, 3, ...\}$ is the set of \textit{natural numbers}. + \item $\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$ is the set of \textit{integers}. + \item $\mathbb{Q}$ is the set of \textit{rational numbers}. + \item $\mathbb{R}$ is the set of \textit{real numbers}. +\end{itemize} +\note[Note]{$\mathbb{Z}$ is called \say{blackboard zee} or \say{big zee.} Naturally, $\mathbb{N}$, $\mathbb{Q}$, and $\mathbb{R}$ have similar names. \\ This, of course, depends on context. Sometimes \say{zee} is all you need.} + + +\problem{} +Which of the following sets contain 100? \par +\hint{There may be more than one answer in all the problems below.} + +\begin{tcolorbox}[ + colback=white, + colframe=black, + width=0.5\textwidth, + toprule=0.3mm, + bottomrule=0.3mm, + leftrule=0.3mm, + rightrule=0.3mm, +] + \hfill $\mathbb{N}$ \hfill $\mathbb{Z}$ \hfill $\mathbb{Q}$ \hfill $\mathbb{R}$ \hfill\null +\end{tcolorbox} +\vfill + + +\problem{} +Which of the following sets contain {\large $\frac{1}{2}$}? \par + +\begin{tcolorbox}[ + colback=white, + colframe=black, + width=0.5\textwidth, + toprule=0.3mm, + bottomrule=0.3mm, + leftrule=0.3mm, + rightrule=0.3mm, +] + \hfill $\mathbb{N}$ \hfill $\mathbb{Z}$ \hfill $\mathbb{Q}$ \hfill $\mathbb{R}$ \hfill\null +\end{tcolorbox} +\vfill + + + +\problem{} +Which of the following sets contain $\pi$? \par + +\begin{tcolorbox}[ + colback=white, + colframe=black, + width=0.5\textwidth, + toprule=0.3mm, + bottomrule=0.3mm, + leftrule=0.3mm, + rightrule=0.3mm, +] + \hfill $\mathbb{N}$ \hfill $\mathbb{Z}$ \hfill $\mathbb{Q}$ \hfill $\mathbb{R}$ \hfill\null +\end{tcolorbox} +\vfill + + +\problem{} +Which of the following sets contain $\sqrt{-1}$? \par + +\begin{tcolorbox}[ + colback=white, + colframe=black, + width=0.5\textwidth, + toprule=0.3mm, + bottomrule=0.3mm, + leftrule=0.3mm, + rightrule=0.3mm, +] + \hfill $\mathbb{N}$ \hfill $\mathbb{Z}$ \hfill $\mathbb{Q}$ \hfill $\mathbb{R}$ \hfill\null +\end{tcolorbox} + +\vfill +\pagebreak + + +\definition{} +Consider the sets $A$ and $B$. The set $A \times B$ consists of all ordered\footnotemark{} pairs $(a, b)$ where $a \in A$ and $b \in B$. \par +This is called the \textit{cartesian product}, and is usually pronounced \say{$A$ cross $B$}. + +\footnotetext{This means that order matters. $(a, b) \neq (b, a)$.} + +\vspace{2mm} + + +For example, $\{1, 2, 3\} \times \{\heartsuit, \star\} = \{(1,\heartsuit),~ (1, \star),~ (2,\heartsuit),~ (2, \star),~ (3,\heartsuit),~ (3, \star)\}$ \par + + +You can think of this as placing the two sets \say{perpendicular} to one another: + +\begin{center} +\begin{tikzpicture}[ + scale=1, + bullet/.style={circle,inner sep=1.5pt,fill} +] + \draw[->] (-0.2,0) -- (4,0) node[right]{$A$}; + \draw[->] (0,-0.2) -- (0,3) node[above]{$B$}; + + \draw (1,0.1) -- ++ (0,-0.2) node[below]{$1$}; + \draw (2,0.1) -- ++ (0,-0.2) node[below]{$2$}; + \draw (3,0.1) -- ++ (0,-0.2) node[below]{$3$}; + + \draw (0.1, 1) -- ++ (-0.2, 0) node[left]{$\heartsuit$}; + \draw (0.1, 2) -- ++ (-0.2, 0) node[left]{$\star$}; + + \node[bullet] at (1, 1){}; + \node[bullet] at (2, 1) {}; + \node[bullet] at (3, 1) {}; + \node[bullet] at (1, 2) {}; + \node[bullet] at (2, 2) {}; + \node[bullet] at (3, 2) {}; + + + \draw[rounded corners] (0.5, 0.5) rectangle (3.5, 2.5) {}; + \node[above] at (2, 2.5) {$A \times B$}; + +\end{tikzpicture} +\end{center} + +\problem{} +Let $A = \{0, 1\} \times \{0, 1\}$ \par +Let $B = \{ a, b\}$ \par +What is $A \times B$? + +\vfill + +\problem{} +What is $\mathbb{R} \times \mathbb{R}$? \par +\hint{Use the \say{perpendicular} analogy} + +\vfill +\pagebreak + +\definition{} +$\mathbb{R}^n$ is the set of $n$-tuples of real numbers. \par +In English, this means that an element of $\mathbb{R}^n$ is a list of $n$ real numbers: \par + +\vspace{4mm} + +Elements of $\mathbb{R}^2$ look like $(a, b)$, where $a, b \in \mathbb{R}$. \hfill \note{\textit{Note:} $\mathbb{R}^2$ is pronounced \say{arrgh-two.}} +Elements of $\mathbb{R}^5$ look like $(a_1, a_2, a_3, a_4, a_5)$, where $a_n \in \mathbb{R}$. \par + +$\mathbb{R}^1$ and $\mathbb{R}$ are identical. + +\vspace{4mm} + +Intuitively, $\mathbb{R}^2$ forms a two-dimensional plane, and $\mathbb{R}^3$ forms a three-dimensional space. \par +$\mathbb{R}^n$ is hard to visualize when $n \geq 4$, but you are welcome to try. + +\problem{} +Convince yourself that $\mathbb{R} \times \mathbb{R}$ is $\mathbb{R}^2$. \par +What is $\mathbb{R}^2 \times \mathbb{R}$? +\vfill + +\problem{} +What is $\mathbb{N}^2$? +\vfill + +\problem{} +What is $\mathbb{Z}^3$? + +\vfill +\pagebreak \ No newline at end of file diff --git a/Advanced/Size of Sets/parts/3 functions.tex b/Advanced/Size of Sets/parts/3 functions.tex new file mode 100644 index 0000000..5f6ac93 --- /dev/null +++ b/Advanced/Size of Sets/parts/3 functions.tex @@ -0,0 +1,378 @@ +\section{Functions and Maps} + +\definition{} +A \textit{function} or \textit{map} $f$ from a set $A$ to a set $B$ is a rule that assigns an element of $B$ to each element of $A$. We write this as $f: A \to B$. + +\vspace{1mm} + +Let $L = \{\texttt{a}, \texttt{b}, \texttt{c}, \texttt{d}, ..., \texttt{z}\}$ be the set of lowercase english letters. \par +Let $C = \{\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}, ..., \texttt{Z}\}$ be the set of uppercase english letters. \par + +\vspace{1mm} + +Say we have a function $g: L \to C$ that capitalizes english letters. \par +We can think of this function as a \textit{map} from $A$ to $B$, shown below using arrows: + + +\begin{center} + \begin{tikzpicture}[scale=0.5] + \node[anchor=south] at (0, 1) {Set $A$}; + \draw[line width = 0.25mm, rounded corners=2mm] (-1, 1) rectangle (1, -5); + \node at (0, 0) {\texttt{a}}; + \node at (0, -1) {\texttt{b}}; + \node at (0, -2) {\texttt{c}}; + \node at (0, -3) {\texttt{d}}; + \node at (0, -4) {\texttt{...}}; + + \draw[line width = 0.4mm, ->, gray] (0.5, 0) -- (3.5, 0); + \draw[line width = 0.4mm, ->, gray] (0.5, -1) -- (3.5, -1); + \draw[line width = 0.4mm, ->, gray] (0.5, -2) -- (3.5, -2); + \draw[line width = 0.4mm, ->, gray] (0.5, -3) -- (3.5, -3); + \node[fill=white, text=gray] at (2, 0) {$g$}; + + \node[anchor=south] at (4, 1) {Set $B$}; + \draw[line width = 0.25mm, rounded corners=2mm] (3, 1) rectangle (5, -5); + \node at (4, 0) {\texttt{A}}; + \node at (4, -1) {\texttt{B}}; + \node at (4, -2) {\texttt{C}}; + \node at (4, -3) {\texttt{D}}; + \node at (4, -4) {\texttt{...}}; + \end{tikzpicture} +\end{center} + + +\definition{} +We say a map $f$ is \textit{one-to-one} if $a = b$ implies $f(a) = f(b)$ for all $a, b \in A$. \par +In other words, this means that no two elements of $A$ are mapped to the same $b$: + +\null\hfill +\begin{minipage}{0.48\textwidth} +\begin{center} +\textbf{A one-to-one function:} \par +\vspace{2mm} +\begin{tikzpicture}[scale=0.5] + \node[anchor=south] at (0, 1) {Set $A$}; + \draw[line width = 0.25mm, rounded corners=2mm] (-1, 1) rectangle (1, -4); + \node at (0, 0) {\texttt{1}}; + \node at (0, -1) {\texttt{2}}; + \node at (0, -2) {\texttt{3}}; + \node at (0, -3) {\texttt{4}}; + + \draw[line width = 0.4mm, ->, gray] (0.5, 0) -- (3.5, 0); + \draw[line width = 0.4mm, ->, gray] (0.5, -2) -- (3.5, -1); + \draw[line width = 0.4mm, ->, gray] (0.5, -1) -- (3.5, -3); + \draw[line width = 0.4mm, ->, gray] (0.5, -3) -- (3.5, -2); + \node[fill=white, text=gray] at (2, 0) {$f$}; + + \node[anchor=south] at (4, 1) {Set $B$}; + \draw[line width = 0.25mm, rounded corners=2mm] (3, 1) rectangle (5, -5); + \node at (4, 0) {\texttt{a}}; + \node at (4, -1) {\texttt{b}}; + \node at (4, -2) {\texttt{c}}; + \node at (4, -3) {\texttt{d}}; + \node at (4, -4) {\texttt{e}}; +\end{tikzpicture} +\end{center} +\end{minipage} +\hfill +\begin{minipage}{0.48\textwidth} +\begin{center} +\textbf{NOT a one-to-one function:} \par +\vspace{2mm} +\begin{tikzpicture}[scale=0.5] + \node[anchor=south] at (0, 1) {Set $A$}; + \draw[line width = 0.25mm, rounded corners=2mm] (-1, 1) rectangle (1, -4); + \node at (0, 0) {\texttt{1}}; + \node at (0, -1) {\texttt{2}}; + \node at (0, -2) {\texttt{3}}; + \node at (0, -3) {\texttt{4}}; + + \draw[line width = 0.4mm, ->, gray] (0.5, 0) -- (3.5, 0); + \draw[line width = 0.4mm, ->, gray] (0.5, -2) -- (3.5, -1); + \draw[line width = 0.4mm, ->, gray] (0.5, -1) -- (3.4, -1.8); + \draw[line width = 0.4mm, ->, gray] (0.5, -3) -- (3.4, -2.2); + \node[fill=white, text=gray] at (2, 0) {$f$}; + + \node[anchor=south] at (4, 1) {Set $B$}; + \draw[line width = 0.25mm, rounded corners=2mm] (3, 1) rectangle (5, -5); + \node at (4, 0) {\texttt{a}}; + \node at (4, -1) {\texttt{b}}; + \node at (4, -2) {\texttt{c}}; + \node at (4, -3) {\texttt{d}}; + \node at (4, -4) {\texttt{e}}; + + \draw[line width = 0.4mm, ->] (5.5, -2) -- (4.5, -2); + \node[anchor=west] at (5.5, -2) {!!!}; + +\end{tikzpicture} +\end{center} +\end{minipage} +\hfill\null + +\vfill + + +\definition{} +We say a map $f$ is \textit{onto} if for every $b \in B$, there is an $a \in A$ so that $b = f(a)$. \par +In other words, this means that every element of $B$ has some element of $A$ mapped to it: + +\null\hfill +\begin{minipage}{0.48\textwidth} +\begin{center} +\textbf{An onto function:} \par +\vspace{2mm} +\begin{tikzpicture}[scale=0.5] + \node[anchor=south] at (0, 1) {Set $A$}; + \draw[line width = 0.25mm, rounded corners=2mm] (-1, 1) rectangle (1, -5); + \node at (0, 0) {\texttt{1}}; + \node at (0, -1) {\texttt{2}}; + \node at (0, -2) {\texttt{3}}; + \node at (0, -3) {\texttt{4}}; + \node at (0, -4) {\texttt{5}}; + + \draw[line width = 0.4mm, ->, gray] (0.5, 0) -- (3.5, 0); + \draw[line width = 0.4mm, ->, gray] (0.5, -1) -- (3.5, -2); + \draw[line width = 0.4mm, ->, gray] (0.5, -2) -- (3.5, -1); + \draw[line width = 0.4mm, ->, gray] (0.5, -3) -- (3.4, -3); + \draw[line width = 0.4mm, ->, gray] (0.5, -4) -- (3.4, -3.4); + \node[fill=white, text=gray] at (2, 0) {$f$}; + + \node[anchor=south] at (4, 1) {Set $B$}; + \draw[line width = 0.25mm, rounded corners=2mm] (3, 1) rectangle (5, -4); + \node at (4, 0) {\texttt{a}}; + \node at (4, -1) {\texttt{b}}; + \node at (4, -2) {\texttt{c}}; + \node at (4, -3.1) {\texttt{d}}; +\end{tikzpicture} +\end{center} +\end{minipage} +\hfill +\begin{minipage}{0.48\textwidth} +\begin{center} +\textbf{NOT an onto function:} \par +\vspace{2mm} +\begin{tikzpicture}[scale=0.5] + \node[anchor=south] at (0, 1) {Set $A$}; + \draw[line width = 0.25mm, rounded corners=2mm] (-1, 1) rectangle (1, -5); + \node at (0, 0) {\texttt{1}}; + \node at (0, -1) {\texttt{2}}; + \node at (0, -2) {\texttt{3}}; + \node at (0, -3) {\texttt{4}}; + \node at (0, -4) {\texttt{5}}; + + \draw[line width = 0.4mm, ->, gray] (0.5, 0) -- (3.5, -0.8); + \draw[line width = 0.4mm, ->, gray] (0.5, -1) -- (3.5, -2); + \draw[line width = 0.4mm, ->, gray] (0.5, -2) -- (3.5, -1.2); + \draw[line width = 0.4mm, ->, gray] (0.5, -3) -- (3.4, -3); + \draw[line width = 0.4mm, ->, gray] (0.5, -4) -- (3.4, -3.4); + \node[fill=white, text=gray] at (2, -0.4) {$f$}; + + \node[anchor=south] at (4, 1) {Set $B$}; + \draw[line width = 0.25mm, rounded corners=2mm] (3, 1) rectangle (5, -4); + \node at (4, 0) {\texttt{a}}; + \node at (4, -1) {\texttt{b}}; + \node at (4, -2) {\texttt{c}}; + \node at (4, -3.1) {\texttt{d}}; + + \draw[line width = 0.4mm, ->] (5.5, 0) -- (4.5, 0); + \node[anchor=west] at (5.5, 0) {!!!}; +\end{tikzpicture} +\end{center} +\end{minipage} +\hfill\null + +\vfill +\pagebreak + +\generic{Remark:} +The words \say{function} and \say{map} are two views of the same mathematical object. We usually think of functions as \say{machines} that take an input, change it, and produce an output. We think of maps as \say{rules} that match each element of a set $A$ to an element of a set $B$. + +\vspace{2mm} + +Again, functions and maps are \textit{identical}. They do the same thing. The only difference between \say{functions} and \say{maps} is how we think about them. + +% one-to-one = injective +% onto = surjective + +\problem{} +Is the \say{capitalize} function in \ref{deffun} one-to-one? Is it onto? + +\vfill + +\problem{} +Consider the function $f: \mathbb{Z} \to \mathbb{Z}$ defined by $f(x) = x^2$. \par +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? + +\[ + f(x) = \begin{cases} + 0 & \text{if } x = 0 \\ + x + 1 & \text{otherwise} + \end{cases} +\] + + +% TODO: +% bijections, same size if exists bijection + +\vfill +\pagebreak + +\definition{Invertible Functions} +A function $g$ is an \textit{inverse} of a function $f$ if $g(f(x)) = x$ for any $x$. \par +In other words, the function $g$ \say{undoes} $f$. Usually, the inverse of a function $f$ is written $f^{-1}$. \par +We say a function is \textit{invertible} if it has an inverse. + +\vspace{2mm} + +Intuitively, we could say that the inverse of $f$ reverses the \say{arrows} of $f$. + +\problem{} +Is the following function invertible? \par +Draw the inverse, or explain why you can't. + +\begin{center} +\begin{tikzpicture}[scale=0.5] + \node[anchor=south] at (0, 1) {$A$}; + \draw[line width = 0.25mm, rounded corners=2mm] (-1, 1) rectangle (1, -4); + \node at (0, 0) {\texttt{1}}; + \node at (0, -1) {\texttt{2}}; + \node at (0, -2) {\texttt{3}}; + \node at (0, -3) {\texttt{4}}; + + \draw[line width = 0.4mm, ->, gray] (0.5, 0) -- (3.5, 0); + \draw[line width = 0.4mm, ->, gray] (0.5, -2) -- (3.5, -1); + \draw[line width = 0.4mm, ->, gray] (0.5, -1) -- (3.5, -3); + \draw[line width = 0.4mm, ->, gray] (0.5, -3) -- (3.5, -2); + \node[fill=white, text=gray] at (2, 0) {$f$}; + + \node[anchor=south] at (4, 1) {$B$}; + \draw[line width = 0.25mm, rounded corners=2mm] (3, 1) rectangle (5, -4); + \node at (4, 0) {\texttt{a}}; + \node at (4, -1) {\texttt{b}}; + \node at (4, -2) {\texttt{c}}; + \node at (4, -3) {\texttt{d}}; +\end{tikzpicture} +\end{center} + +\vfill + +\problem{} +Is the following function invertible? \par +Draw the inverse, or explain why you can't. + +\begin{center} +\begin{tikzpicture}[scale=0.5] + \node[anchor=south] at (0, 1) {Set $A$}; + \draw[line width = 0.25mm, rounded corners=2mm] (-1, 1) rectangle (1, -4); + \node at (0, 0) {\texttt{1}}; + \node at (0, -1) {\texttt{2}}; + \node at (0, -2) {\texttt{3}}; + \node at (0, -3) {\texttt{4}}; + + \draw[line width = 0.4mm, ->, gray] (0.5, 0) -- (3.5, 0); + \draw[line width = 0.4mm, ->, gray] (0.5, -1) -- (3.5, -1); + \draw[line width = 0.4mm, ->, gray] (0.5, -2) -- (3.5, -3); + \draw[line width = 0.4mm, ->, gray] (0.5, -3) -- (3.5, -2); + \node[fill=white, text=gray] at (2, 0) {$f$}; + + \node[anchor=south] at (4, 1) {Set $B$}; + \draw[line width = 0.25mm, rounded corners=2mm] (3, 1) rectangle (5, -5); + \node at (4, 0) {\texttt{a}}; + \node at (4, -1) {\texttt{b}}; + \node at (4, -2) {\texttt{c}}; + \node at (4, -3) {\texttt{d}}; + \node at (4, -4) {\texttt{e}}; +\end{tikzpicture} +\end{center} + +\vfill + + +\problem{} +Is the following function invertible? \par +Draw the inverse, or explain why you can't. + +\begin{center} +\begin{tikzpicture}[scale=0.5] + \node[anchor=south] at (0, 1) {Set $A$}; + \draw[line width = 0.25mm, rounded corners=2mm] (-1, 1) rectangle (1, -5); + \node at (0, 0) {\texttt{1}}; + \node at (0, -1) {\texttt{2}}; + \node at (0, -2) {\texttt{3}}; + \node at (0, -3) {\texttt{4}}; + \node at (0, -4) {\texttt{5}}; + + \draw[line width = 0.4mm, ->, gray] (0.5, 0) -- (3.5, 0); + \draw[line width = 0.4mm, ->, gray] (0.5, -1) -- (3.5, -2); + \draw[line width = 0.4mm, ->, gray] (0.5, -2) -- (3.5, -1); + \draw[line width = 0.4mm, ->, gray] (0.5, -3) -- (3.4, -3); + \draw[line width = 0.4mm, ->, gray] (0.5, -4) -- (3.4, -3.4); + \node[fill=white, text=gray] at (2, 0) {$f$}; + + \node[anchor=south] at (4, 1) {Set $B$}; + \draw[line width = 0.25mm, rounded corners=2mm] (3, 1) rectangle (5, -4); + \node at (4, 0) {\texttt{a}}; + \node at (4, -1) {\texttt{b}}; + \node at (4, -2) {\texttt{c}}; + \node at (4, -3.1) {\texttt{d}}; +\end{tikzpicture} +\end{center} + +\vfill +\pagebreak + + +\definition{Bijections} +One-to-one maps are also called \textit{injective} maps. \par +Onto maps are also called \textit{surjective} maps. + +\vspace{2mm} + +If a function is both one-to-one and onto, we say it is a \textit{bijection}. + +\vspace{4mm} + +\theorem{} +All bijective functions are invertible. All invertible functions are bijections. \par +You should review the problems on the previous page and convince yourself that this is true. + +\problem{} +We say a set $S$ is \textit{finite} if there exists a bijection from $S$ to $\{1, 2, 3, ..., n\}$ for some integer $n$.\par +Convince yourself that this definition of \say{finite-ness} is the same as the one in \ref{infiniteset}. + +\problem{} +Is there a bijection between the sets $\{1, 2, 3\}$ and $\{\texttt{A}, \texttt{B}, \texttt{C}\}$? \par +If a bijection exists, find one; if one doesn't, prove it. \par +\vfill + +\problem{} +Is there a bijection between the sets $\{1, 2, 3, 4\}$ and $\{\texttt{A}, \texttt{B}, \texttt{C}\}$? \par +If a bijection exists, find one; if one doesn't, prove it. \par +\vfill + + +\problem{} +Let $A$ and $B$ be two sets of different sizes. \par +Show that no bijection between $A$ and $B$ exists. +\vfill + + +\ref{samesize} reveals a very important fact: if we can find a bijection between two sets $A$ and $B$, these sets must have the same number of elements. Similarly, if we know that a bijection doesn't exist, we know that $A$ and $B$ must have a different number of elements. + +\vspace{2mm} + +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. + +\pagebreak diff --git a/Advanced/Size of Sets/parts/4 dense.tex b/Advanced/Size of Sets/parts/4 dense.tex new file mode 100644 index 0000000..8d59e69 --- /dev/null +++ b/Advanced/Size of Sets/parts/4 dense.tex @@ -0,0 +1,35 @@ +\section{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 +A few examples are below: +\begin{itemize} + \item $\mathbb{Z}$ is an ordered set under $<$. + \item $\{\texttt{A}, \texttt{B}, ..., \texttt{Z}\}$ is an ordered set under $\diamond$,\par + Where $\alpha \diamond \beta$ holds iff the letter $\alpha$ comes before letter $\beta$ in the alphabet. +\end{itemize} + + +\definition{} +We say an ordered set $A$ is \textit{dense} if for any $a, b \in A$ there is a $c \in A$ so that $a < c < b$.\par +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. +\vfill + +\problem{} +Show that the ordered set $(\mathbb{R}, <)$ is dense. +\vfill + +\problem{} +Show that there is a real number between every two rationals. +\vfill + +\problem{} +Show that there is a rational number between every two reals. +\vfill + +\pagebreak \ No newline at end of file