61 lines
1.3 KiB
TeX
61 lines
1.3 KiB
TeX
|
% 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}
|