# One element set

As **singleton** , **elementary quantity** , **of an amount** or (English) **Singleton** in the are mathematics those amounts indicated that exactly one element included. A set is therefore one-element, exactly when it has the power of one. For example, there is a one-element set, but also , because here the only element is the set (which in turn is not one-element).

The existence of one-element sets follows in the Zermelo-Fraenkel set theory from the pair set axiom , which says that there is for sets and also set. If you choose , so is . The existence of the one-element set containing the empty set follows using the empty set axiom .

In von Neumann's model of natural numbers , every natural number contains exactly elements, so the only one-element number is .

If is an arbitrary set and is a one-element set, then there is exactly one function of after , namely . Thus, the set of all functions of by , also a singleton.

In the category of sets, singlesets are terminal objects and are isomorphic to one another. The last statement in the previous paragraph can therefore be formulated there as the simple equation .

## Equivalences

is an element of if and only if .

and have empty cut if and only if not equal if and only if not equal .

exactly when .