Disjoint union (set)

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Note: a closely related concept is that of a tagged union

Definition

Let $(X_\alpha)_{\alpha\in I}$ be an arbitrary family of sets. We denote their disjoint union or coproduct as $\coprod_{\alpha\in I}X_\alpha$ and we define this to be:

• $(\beta,x)\in\coprod_{\alpha\in I}X_\alpha\iff(\beta\in I\wedge x\in X_\beta)$
• We could also define $\coprod_{\alpha\in I}X_\alpha$ as sets of the form $(x,\beta)$ instead. It doesn't matter.

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TODO: Construction as a set

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With this we get canonical injections, let $\beta\in I$ be given, then:

• $i_\beta:X_\beta\rightarrow\coprod_{\alpha\in I}X_\alpha$ given by $i_\beta:x\mapsto(\beta,x)$

It is common to identify $X_\alpha$ with its image, $i_\alpha(X_\alpha)$, or to define $X_\beta^*:=i_\beta(X_\beta)$

See also

• Cartesian product
• Product of sets
• Canonical injections
  • Canonical injections of a disjoint union
• Coproduct of sets