Singleton (mathematics)

As a singleton, singleton or (english) Singleton those amounts referred to in the mathematics that contain exactly one element. A set is a singleton that is, if and only if it has the power one. For example, {1 } is a singleton, but also because here the only element of the set { 1,2,3} (which in turn is not a singleton ).

The existence of one-element sets follows in the Zermelo -Fraenkel set theory of the pair axiom, which states that for quantities x and y and { x, y} is set. If we choose x = y, then { x, y} = { x, x} = { x}. The existence of the singleton set containing the empty set, in this case using the following axiom empty set.

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

If X is an arbitrary set and is a singleton, then there exists a function from X to A, namely. Thus the set of all functions from X to A, also a singleton.

In the category of sets singletons are terminal objects and mutually isomorphic. The last statement in the previous paragraph can be formulated as the simple equation so there.

Equivalences

X is an element of { a} if and only if x = a

{a } and { b} have empty intersection if and only if a not = b if and only if { a} than { b}.

{a } = { b} if and only if a = b.

• Set theory