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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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