Disjoint union

In mathematics, the disjoint union of two sets X and Y is a set which contains disjoint (that is, non-intersecting) "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 Xλ as λ ranges over a general index set Λ may be defined as


 * $$\coprod_{\lambda \in \Lambda} X_\lambda = \bigcup_{\lambda \in \Lambda} \{\lambda\} \times X_\lambda . \, $$