# Countable Set **A [[set]] for which there exists a [[Bijective Function|one-to-one correspondance]] with the set of natural numbers.** ![[CountableSet.svg|500]] *The set of integers is a countably infinite set; surprisingly, there are equally as many integers as there are natural numbers* > [!NOTE] Countable Set > A set $S$ is *countable* if any of the following are true: > - the [[cardinality]] $|S|$ is less than or equal to $\aleph_{0}$, the cardinality of the set of natural numbers $\mathbb{N}$. > - there exists an [[Injective Function|injective]] function from $S$ to $\mathbb{N}$; there exists a [[Surjective Function|surjective]] function from $\mathbb{N}$ to $S$. A set is *countable* if every element within the set can be assigned a unique natural number. A countable set may be finite or infinite, and an infinite countable set is known as *countably infinite*.