# Definition:Singleton

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## 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.

## Sources

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