# 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 . ${\ displaystyle a = b}$

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