Disjoint union (set)

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

Definition

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

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

TODO: Construction as a set

With this we get canonical injections, let [ilmath]\beta\in I[/ilmath] be given, then:

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

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