Types of Sets I — Empty, Finite, Infinite, Singleton and Universal Sets

A handful of set types recur often enough to deserve their own names.

Empty set (null set): a set with no elements at all, written $\emptyset$ or $\{\ \}$. For example, $\{x : x \in \mathbb{N}, x + 2 = 1\} = \emptyset$, since no natural number satisfies $x = -1$. Note carefully that $\{0\}$ is not empty — it is a set holding exactly one element, the number $0$.

Singleton set: a set with exactly one element, such as $\{9\}$ or $\{x : x \in \mathbb{N}, x^2 = 64\} = \{8\}$.

Finite set: a set whose elements can be counted completely, the count ending at some definite whole number. $\{3, 6, 9, 12\}$ is finite (4 elements); so, technically, is $\emptyset$ (0 elements). A set whose counting process never terminates — such as $\mathbb{N}$ or $\mathbb{Z}$ — is called an infinite set.

Universal set: the set, usually written $U$, containing every object relevant to the particular problem under discussion; every other set in that discussion is understood to be drawn from within $U$. If a shop stocks sarees in six colours, $U$ could be taken as the set of all six colours it stocks, and "colours currently out of stock" would then be some subset of that $U$.