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). ${\displaystyle \{1\}}$ ${\displaystyle \{\{1,2,3\}\}}$ ${\displaystyle \{1,2,3\}}$

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 . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \{x,y\}}$ ${\displaystyle x=y}$ ${\displaystyle \{x,y\}=\{x,x\}=\{x\}}$

In von Neumann's model of natural numbers , every natural number contains exactly elements, so the only one-element number is . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle 1:=\{\emptyset \}}$

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. ${\displaystyle X}$ ${\displaystyle A=\{a\}}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle f\colon X\to A,x\mapsto a}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle A^{X}}$

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 . ${\displaystyle 1^{X}\simeq 1}$

Equivalences

${\displaystyle x}$is an element of if and only if . ${\displaystyle \{a\}}$ ${\displaystyle x=a}$

${\displaystyle x\in \{a\}\Longleftrightarrow x=a}$

${\displaystyle \{a\}}$and have empty cut if and only if not equal if and only if not equal . ${\displaystyle \{b\}}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \{a\}}$ ${\displaystyle \{b\}}$

${\displaystyle \{a\}\cap \{b\}=\emptyset \Longleftrightarrow a\neq b\Longleftrightarrow \{a\}\neq \{b\}}$

${\displaystyle \{a\}=\{b\}}$exactly when . ${\displaystyle a=b}$

${\displaystyle \{a\}=\{b\}\Longleftrightarrow a=b}$