Difference between revisions of "Disjoint union (set)"
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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: | 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)}} | * {{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}} | {{Todo|Construction as a set}} | ||
With this we get ''canonical injections'', let {{M|\beta\in I}} be given, then: | With this we get ''canonical injections'', let {{M|\beta\in I}} be given, then: | ||
Latest revision as of 20:21, 25 September 2016
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- Note: a closely related concept is that of a tagged union
Contents
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]
