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{\bf MAT 324, Fall 2011} \\
{\bf  PROBLEM SET 1}
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\begin{enumerate}
\item Let $X$ be the set of all real roots of all polynomials with 
      integer coefficients. Is $X$ countable or uncountable?
      Explain why.

\item  Define  $x \sim y$ if $x-y$ is rational.  Prove this is an 
       equivalence relation on the reals. How many distinct equivalence
       classes are there?

\item  Is the function $f(x) = \sum_{n=0}^\infty 2^{-n} \sin(2^n x)$ 
       Riemann integrable on $[0, 2 \pi]$?  Explain why or why not.

\item  Is there a compact, uncountable set of real numbers which contains 
       no rational numbers?  Give an example or prove no such set exists.

\item What is the average distance between two random points in $[0,1]$?
      We have not had enough theory yet to make this precise, but see 
      if you can come up with a plausible number and explanation for it.

\end{enumerate}

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