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.


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Also known as

Some authors refer to a singleton as a unit set.

Some refer to it as a one-element set.

Some use the phrase singleton set.


Also see

  • Results about singletons can be found here.


Sources

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