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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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    \singlespacing

  \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
%      its value: for instance, ``the positive square rrot of two'' and
%      ``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}

\end{document}