What are some examples of countable infinity?

What are some examples of countable infinity?

Countable infinity refers to sets that can be put into a one-to-one correspondence with the natural numbers. Examples include the set of natural numbers, integers, and rational numbers.

Can you explain why the set of rational numbers is countably infinite?

Sure! The set of rational numbers can be listed in a sequence, even though they are dense in the real numbers. We can arrange them in a grid and use a diagonal argument to show that we can enumerate them.

What about the set of even numbers? Is that also countably infinite?

Yes, the set of even numbers is countably infinite. You can map each even number to a natural number by dividing it by 2, creating a one-to-one correspondence.

Are there any other interesting properties of countably infinite sets?

Absolutely! One interesting property is that the union of a finite number of countably infinite sets is also countably infinite. Additionally, any subset of a countably infinite set is either finite or countably infinite.