View source for Disjoint union (set)

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: '''Note: ''' a closely related concept is that of a [[tagged union]]
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==Definition==
Let {{M|(X_\alpha)_{\alpha\in I} }} be an arbitrary family of [[sets]]. We denote their ''disjoint union'' or ''{{link|coproduct|category theory}}'' as {{M|1=\coprod_{\alpha\in I}X_\alpha}} and we define this to be:
* {{M|1=(\beta,x)\in\coprod_{\alpha\in I}X_\alpha\iff(\beta\in I\wedge x\in X_\beta)}}
* We could also define {{M|\coprod_{\alpha\in I}X_\alpha}} as sets of the form {{M|(x,\beta)}} instead. It doesn't matter. 
{{Todo|Construction as a set}}
With this we get ''canonical injections'', let {{M|\beta\in I}} be given, then:
* {{M|i_\beta:X_\beta\rightarrow\coprod_{\alpha\in I}X_\alpha}} given by {{M|i_\beta:x\mapsto(\beta,x)}}
It is common to identify {{M|X_\alpha}} with its image, {{M|i_\alpha(X_\alpha)}}, or to define {{M|1=X_\beta^*:=i_\beta(X_\beta)}}
==See also==
* [[Cartesian product]]
* [[Product of sets]]
* [[Canonical injections]]
** [[Canonical injections of a disjoint union]]
* [[Coproduct of sets]]
==References==
<references/>
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{{Definition|Set Theory|Elementary Set Theory}}