From 0afeaa8c6d3865e58b81736e481e02096f85bfc0 Mon Sep 17 00:00:00 2001 From: Mark Date: Wed, 19 Jul 2023 09:55:30 -0700 Subject: [PATCH] Edits --- Advanced/Size of Sets/main.tex | 55 +++++++++---------- Advanced/Size of Sets/parts/3 functions.tex | 9 +-- Advanced/Size of Sets/parts/4 enumeration.tex | 46 ++++++++++++++++ .../parts/{4 dense.tex => dense.tex} | 10 ++-- 4 files changed, 81 insertions(+), 39 deletions(-) create mode 100644 Advanced/Size of Sets/parts/4 enumeration.tex rename Advanced/Size of Sets/parts/{4 dense.tex => dense.tex} (71%) diff --git a/Advanced/Size of Sets/main.tex b/Advanced/Size of Sets/main.tex index f9208aa..557212a 100755 --- a/Advanced/Size of Sets/main.tex +++ b/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 - \pagebreak + %\vfill + %\pagebreak - \section{Bonus Problems} + %\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, $(a, b) \neq (b, a)$. + %\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} + %\begin{solution} + % $(a, b) = \{ \{a\}, \{a, b\}\}$ + %\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{} - 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 one-to-one. Must $h(x) = g(f(x))$ be one-to-one? \par - Provide a proof or a counterexample. + %\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 - - \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 + %\vfill + %\pagebreak \end{document} \ 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 index 5f6ac93..9b8bc18 100644 --- a/Advanced/Size of Sets/parts/3 functions.tex +++ b/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 diff --git a/Advanced/Size of Sets/parts/4 enumeration.tex b/Advanced/Size of Sets/parts/4 enumeration.tex new file mode 100644 index 0000000..bf5d76f --- /dev/null +++ b/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 \ No newline at end of file diff --git a/Advanced/Size of Sets/parts/4 dense.tex b/Advanced/Size of Sets/parts/dense.tex similarity index 71% rename from Advanced/Size of Sets/parts/4 dense.tex rename to Advanced/Size of Sets/parts/dense.tex index 8d59e69..b3fbb87 100644 --- a/Advanced/Size of Sets/parts/4 dense.tex +++ b/Advanced/Size of Sets/parts/dense.tex @@ -1,6 +1,4 @@ -\section{Dense Orderings} - -\note[Note]{This section is fairly difficult. If you get stuck here, try the bonus problems on the next page.} +\section*{Bonus: Dense Orderings} \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{}