Definition:Singleton

Definition

A singleton is a set that contains exactly one element.

The singleton containing only the element $a$ can be written $\set a$.

The set $\set a$ is known as the singleton of $a$.

Warning

It is important to distinguish between an element, for example $a$, and a singleton containing it, that is, $\set a$.

That is $a$ and $\set a$ are not the same thing.

While it is true that:

$a \in \set a$

it is not true that:

$a = \set a$

neither is it true that:

$a \in a$

Formal Definition

The concept of the singleton set can be formalized rigorously as:

$\set A := \set {x : x = A}$

With this definition, the singleton of proper classes is equal to the empty set.

Also known as

Some authors use the term unit set.

Also see

• Results about singletons can be found here.