There are many other ways to construct a bijective mapping from A × B and. It follows from the above that the Cartesian product N × N is countably infinite, that is, | N × N | = ℵ 0 . This result can be generalized to the product of any finite number of countable sets.

What is considered countably infinite?

A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. In other words, one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.

How do you prove countably infinite?

We say a set X is countably infinite if |X| = |N|. If X is infinite, but it is not countably infinite, we say that X is uncountably infinite, or just uncountable. A set X is called countable if it is either finite or countably infinite.