# Disjoint union

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In mathematics, the disjoint union of two sets X and Y is a set which contains "copies" of each of X and Y: it is denoted ${\displaystyle X\amalg Y}$ or, less often, ${\displaystyle X\uplus Y}$.

There are injection maps in1 and in2 from X and Y to the disjoint union, which are injective functions with disjoint images.

If X and Y are disjoint, then the usual union is also a disjoint union. In general, the disjoint union can be realised in a number of ways, for example as

${\displaystyle X\amalg Y=\{0\}\times X\cup \{1\}\times Y.\,}$

The disjoint union has a universal property: if there is a set Z with maps ${\displaystyle f:X\rightarrow Z}$ and ${\displaystyle g:Y\rightarrow Z}$, then there is a map ${\displaystyle h:X\amalg Y\rightarrow Z}$ such that the compositions ${\displaystyle \mathrm {in} _{1}\cdot h=f}$ and ${\displaystyle \mathrm {in} _{2}\cdot h=g}$.

The disjoint union is commutative, in the sense that there is a natural bijection between ${\displaystyle X\amalg Y}$ and ${\displaystyle Y\amalg X}$; it is associative again in the sense that there is a natural bijection between ${\displaystyle X\amalg (Y\amalg Z)}$ and ${\displaystyle (X\amalg Y)\amalg Z}$.

## General unions

The disjoint union of any finite number of sets may be defined inductively, as

${\displaystyle \coprod _{i=1}^{n}X_{i}=X_{1}\amalg (X_{2}\amalg (X_{3}\amalg (\cdots X_{n})\cdots ))).\,}$

The disjoint union of a general family of sets Xλ as λ ranges over a general index set Λ may be defined as

${\displaystyle \coprod _{\lambda \in \Lambda }X_{\lambda }=\bigcup _{\lambda \in \Lambda }\{\lambda \}\times X_{\lambda }.\,}$