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Homework 3

Notes

• Syntax: make sure you’re doing well-defined things with mathematical objects. E.g. it doesn’t a priori make sense to add a set and a number, or a set and a polynomial, or take the union of a polynomial and a set.

• To show a set \(X\) is countable, you need a countable set \(S\) (e.g. \(S = {\mathbb{N}}\)) an either an injection \(X\hookrightarrow S\) or a surjection \(S \twoheadrightarrow X\).

  • Note that any old map won’t suffice: the existence of the inclusion map \({\mathbf{Q}}\hookrightarrow{\mathbf{R}}\) doesn’t show that \({\mathbf{R}}\) is countable.
  • To avoid mistakes, it’s good to try to explicitly write out the sets and say what the map does to elements.

• If you say something is an injection/surjection/bijection, it usually warrants a proof (sometimes short). Common mistake: describing a map and saying it’s a one-to-one correspondence without much or any justification.

• Note that a countable union of countable sets is again countable, but not an arbitrary union. Counterexample: For each \(\alpha\in {\mathbf{R}}\) define \(B_5(\alpha) = \left\{{x\in {\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\lvert {x-\alpha} \right\rvert} < 5}\right\} \cap{\mathbf{Z}}\). Then \({\left\lvert {B_5(\alpha)} \right\rvert} \leq 11\) for any \(\alpha\), so it’s in fact finite, but \(\cup_{\alpha\in {\mathbf{R}}} B_5(\alpha) = {\mathbf{R}}\) is uncountable.

• If saying something is e.g. a countable union of countable sets, try to write out the union, e.g. \(A = \bigcup_{n\in {\mathbb{N}}} A_n\).