What are countable and uncountable sets?

Countable and uncountable sets help us organize things. Suppose we have a collection of toys. Countable sets are like groups of toys we can count, whether we have a few or a lot. On the other hand, uncountable sets are like magic toy boxes that hold so many toys we can’t even count them all!

Learning about countable and uncountable sets helps us make sense of how many things we have and how big collections can be.

Countable set

A set is called countable if it’s finite, like the set of positive whole numbers (like 1, 2, 3, ...). For example, any group of things we can count is a countable set.

Note: Regarding infinite sets, the challenge is figuring out if we can match each element in the set with a positive whole number. If we can, then the infinite set is also countable.

Real-life example: Countable set

Let’s take a collection of identical books in a library. Each book in the collection can be associated with a unique positive whole number, such as 1, 2, 3, and so on, whether the collection is large or small. We can establish a one-to-one correspondence between the books and positive whole numbers. It qualifies as a countable set.

This illustrates that countable sets include finite collections, like a small library, and infinite collections, like an endless series of books.

Uncountable set

If we can’t count all the things in a set, we call it an uncountable set. To prove that a set is uncountable, we have to show that we can’t match up its elements with positive whole numbers (like 1, 2, 3,...).

Real-life example: Uncountable set

An uncountable set is the set of all possible real numbers between 0 and 1. This includes numbers with infinite decimal expansions, for example, 0.123456... or $π/4$. No matter how many real numbers we try to count or write down. There will always be more between 0 and 1 that we haven’t covered. The set is too vast to be paired up with positive whole numbers in a one-to-one manner, making it an uncountable set.

Let’s solve the quiz below to assess our understanding of countable and uncountable sets: