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Advanced/Size of Sets/main.tex
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@ -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}