% 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}