Disjoint union

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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 . \,


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