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 $\left\{{a}\right\}$.

When this happens, we must be careful to distinguish between the element itself, that is, $a$, and the set containing it, that is, $\left\{{a}\right\}$.


The set $\left\{{a}\right\}$ is known as the singleton of $a$.


Formal Definition

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

$\left\{{A}\right\} := \left\{{x : x = A}\right\}$


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