61 lines
1.3 KiB
TeX
Executable File
61 lines
1.3 KiB
TeX
Executable File
% use [nosolutions] flag to hide solutions.
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% use [solutions] flag to show solutions.
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\documentclass[
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solutions,
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singlenumbering
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]{../../resources/ormc_handout}
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\uptitlel{Advanced 1}
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\uptitler{Summer 2023}
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\title{The Size of Sets, Part 1}
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\subtitle{Prepared by Mark on \today{}}
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\begin{document}
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\maketitle
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\input{parts/0 sets.tex}
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\input{parts/1 really big.tex}
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\input{parts/2 cartesian.tex}
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\input{parts/3 functions.tex}
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\input{parts/4 dense.tex}
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\vfill
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\pagebreak
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\section{Bonus Problems}
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\problem{}
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Using only sets, how can we build an ordered pair $(a, b)$? \par
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$(a, b)$ should be equal to $(c, d)$ if and only if $a = b$ and $c = d$. \par
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Of course, $(a, b) \neq (b, a)$.
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\begin{solution}
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$(a, b) = \{ \{a\}, \{a, b\}\}$
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\end{solution}
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\vfill
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\problem{}
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Let $R$ be the set of all sets that do not contain themselves. \par
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Does $R$ exist? \par
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\hint{If $R$ exists, do we get a contradiction?}
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\vfill
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\problem{}
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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
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Provide a proof or a counterexample.
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\vfill
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\problem{}
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Suppose $f: A \to B$ and $g: B \to C$ are both onto. Must $h(x) = g(f(x))$ be onto? \par
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Provide a proof or a counterexample.
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\vfill
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\pagebreak
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\end{document} |