MAT 324, Fall 2006 PROBLEM SET 1

1. Let X be the set of all real roots of all polynomials with integer coefficients. Is X countable or uncountable? Explain why.

2. Define x ∼ y if x − y is rational. Prove this is an equivalence relation on the reals. How many distinct equvalence classes are there?

3. Is the function f(x) = ∑_{n=0}^{∞} 2^{−n} sin(2^{n}x) Riemann integrable on [0, 2π]? Explain why or why not.

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

5. 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 explaination for it.