Rob Bayer
Math 55 Worksheet
June 30, 2009
Instructions
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Introduce yourselves! Despite popular belief, math is in fact a team sport!
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Find some blackboard space, a piece of chalk, and decide who will be your first scribe.
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Do the problems below, having a different person be the scribe for each one.
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Try to work out the problems as a group, but feel free to flag me down if you run into a wall.
Cardinality and Countability
1. Show that the union of two countable sets is countable.
2. Show that the union of countably many countable sets is itself countable. Hint: don’t try to write down an
explicit bijection. Instead, show there is a natural way to enumerate all the elements in the union.
3.
(a) Show that the set of all real numbers that are roots of polynomials with integer coefficients is countable.
(b) Show the same thing, but for polys with rational coefficients. (FYI, such reals are called “algebraic”)
(c) Use part (b) to show that there are uncountably many real numbers that are not algebraic. (FYI, such
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 Summer '08
 STRAIN
 Math, Rational number, Countable set, Georg Cantor

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