# 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.