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\lhead{MATH 311-02}
\chead{Problem Set \#6}
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\rfoot{\bf due April 25}
\begin{document}
\begin{enumerate}
\item \textbf{(24 points)} \emph{Demonstrate the existence of
bijections between the following pairs of sets; you do not need to
explicitly construct the bijection (although in some cases doing
so may be the eaisest approach), but you must appeal to some line
of argumentation that asserts a bijection exists (e.g. the
Cantor-Schr\"oder-Bernstein Theorem).}
\begin{enumerate}
\item \textbf{(4 points)} \emph{The set $\mathbb Z$ and the set of
positive even integers $\{2,4,8,16,\ldots\}$.}
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\item \textbf{(5 points)} \emph{The set $\mathbb N$ and the set of
quadratic functions with integer coefficients
$\{ax^2+bx+c:a,b,c\in\mathbb Z\}$}
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\item \textbf{(5 points)} \emph{The set $\mathbb N$ and the set of
all \emph{finite} subsets of $\mathbb N$.}
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\item \textbf{(5 points)} \emph{The set $\mathbb R$ and the closed
interval $[0,1]$.}
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\item \textbf{(5 points)} \emph{The half-open interval $[0,1)$ and the set
$[0,1)\times[0,1)$.}
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\end{enumerate}
\item \textbf{(6 points)} \emph{A real number is called \emph{transcendental} if it is not a root of any polynomial with integer coefficients. Prove that the set of transcendental numbers is uncountable.}
\item \textbf{(6 points)} \emph{Prove that if $S$ and $T$ are
denumerable sets, so is $S\times T$.}
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\item \textbf{(4 points)} \emph{Show that given a function $S$ and
injective function $f:\mathcal P(S)\rightarrow\mathbb N$, $S$ must
be finite.}
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% \item \textbf{(4 point bonus)} \emph{Let a ``description'' of a
% number be a finite string of letters that uniquely determines
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% ``the positive root of the polynomial x squared minus two'' are
% both descriptions for $\sqrt 2$, and ``the ratio of the
% circumference of a circle to its diameter'' is a description for
% $\pi$. Prove that almost all real numbers do not have
% descriptions.}
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\end{enumerate}
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