Finite, countably infinite and uncountable sets I am having trouble deciding if these equation are finite, countably infinite and uncountable.

*

*$\{ 1, 2, 3, 4, 5 \}$ is countable


*$\{ 2^n \;\vert\; n \in \mathbb{N} \}$ is finite


*$\{ (a,b) \in \mathbb{R} \times \mathbb{R} \;\vert\; a \neq b \}$ is countable


*$ \mathbb{Q} \times \mathbb{Q} \times \mathbb{Q}$ is uncountable (where $\mathbb{Q}$ is the set of the rational numbers)


*$\mathbb{Z}$ is uncountable (where $\mathbb{Z}$ is the set of all integer numbers)


*The cardinality of $\mathbb{Z} \times \mathbb{Q}$ is $\aleph_0$


*$ \{ 2a \;\vert\; a \in \mathbb{Z} \}$ is countably infinite (where $\mathbb{Z}$ is the set of all integer numbers)


*$ \mathbb{Q}^- \cup \mathbb{Z}^- \cup \{ 1, 2, 3 \} $ is countable
Is there anyone who can confirm if I have understod the meaning of finite, countably infinite and uncountable correctly?
 A: A set $S$ is finite if the cardinality of the set (the number of elements in the set), denoted $|S|\in\mathbb{N}$.
Example:
A set you gave was $S = \{1, 2, 3, 4, 5\}$, and clearly $|S| = 5\in\mathbb{N}$ so it is finite.

A set is countable if you can form a bijection (one-to-one correspondence) between the elements of the set and a subset of the natural numbers. In other words, you can enumerate elements in the set.
Example 1: If we again consider $S = \{1, 2, 3, 4, 5\}$, this itself is a subset of $\mathbb{N}$ so by definition it is countable.
Example 2: Whilst your example of $S = \{2^n : n\in\mathbb{N}\}$ is not actually finite (try to see why that is based on the definition), it is countable. This is easy to see as for any $n$, it maps to $2^n$, so there is a one-to-one correspondence between $\mathbb{N}$ and $S$.
Example 3: A cartesian product of finitely many countable sets is countable. $\mathbb{Q}$ is actually countable - there are clever ways to enumerate the rationals (perhaps think about this yourself or look up some examples), so your example of $S=\mathbb{Q}^3$ is actually countable.
Sidenote: We usually define the size of the set of natural numbers, $|\mathbb{N}|=\aleph_0$

For a set $S$ to be uncountable, a few different criteria (but there are more):

*

*There is not a bijection from $S$ to $\mathbb{N}$

*$|S| > |\mathbb{N}|=\aleph_0$

*If $S$ contains a subset which is itself uncountable

Example 1: $\mathbb{Z}$ is countable, but $\mathbb{R}$ is uncountable.
Example 2: $\mathbb{R}\subset\mathbb{C}$, so $\mathbb{C}$ is also uncountable since $\mathbb{R}$ is uncountable.

Exercise: Try and convince yourself / prove for yourself that $\mathbb{Z}$ is countable (but not finite).
