1.11 Infinite Sets

Countable and Uncountable Sets

Which of the following sets are countable?

Exercise 1

\(\mathbb{N} \) is the most basic amongst countably infinite sets.

The set of all integers, \(\mathbb{Z} \), is also countably infinite because integers can be listed in order, so there is a bijection from \(\mathbb{N} \) to \(\mathbb{Z} \) (if you cannot write down the bijection formula right now, go over the reading again!).

The products and quotients of two countably infinite sets are also countably infinite sets, so \(\mathbb{N} \times \mathbb{N} \) and \(\mathbb{Q} \) are countable.

\(\mathbb{Z} ^+ \) is also countable because it is the positive half of a countably infinite set.

The set \(\{0,1\}^{10^{10}} \) is finite with \(2^{10^{10}} \) elements, so it is countable, but \(\{0,1\}^{\omega} \) is not only not finite but also bij \(\text{pow}(\mathbb{N}) \), which is uncountable.

There is a surjection from \(\mathbb{C} \) to \(\mathbb{R} \) to \(\{0,1\}^{\omega} \), by taking the binary expansion of real numbers, so these are also not countable.