Disjoint union

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 $X\amalg Y$ or, less often, $X\uplus Y$.

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

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

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

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

General unions

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

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

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

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

References

• Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold, 24. 
• Keith J. Devlin (1979). Fundamentals of Contemporary Set Theory. Springer-Verlag, 12. ISBN 0-387-90441-7.