View source for Disjoint union

{{TEX|done}}{{MSC|03E}}

''discriminated union'', ''sum''

A construction in set theory corresponding to the [[coproduct]], the union of disjoint "copies" of sets in a family.  Let $X_\lambda$ be a family of sets indexed by $\lambda \in \Lambda$.  The disjoint union $Y = \coprod_{\lambda \in \Lambda} X_\lambda$ has a [[universal property]]: there are maps $i_\lambda : X_\lambda \rightarrow Y$ such that for any family of maps $f_\lambda : X_\lambda \rightarrow Z$ for some $Z$ and all $\lambda \in \Lambda$, there is a map $F : \coprod_{\lambda \in \Lambda} X_\lambda \rightarrow Z$ such that $i_\lambda \circ F = f_\lambda$.

If the $X_\lambda$ are mutually disjoint, so that $\lambda \neq \mu \Rightarrow X_\lambda \cap X_\mu = \emptyset$, then their union $Y = \bigcup_{\lambda \in \Lambda} X_\lambda$ is said to be the (internal) disjoint union of the $X_\lambda$: one also says that the $X_\lambda$ form a [[partition]] or [[decomposition]] of $Y$.  The $i_\lambda$ are the inclusion maps of the $X_\lambda$ into $Y$.

More generally, we may construct a disjoint union given any family $X_\lambda$ as follows.  Let $Y' = \bigcup_{\lambda \in \Lambda} X_\lambda$ and define maps $i_\lambda : X_\lambda \rightarrow Y' \times \Lambda$ by $i_\lambda : x \mapsto (x,\lambda)$.  Then each $i_\lambda$ is an [[injection]], the images of the $i_\lambda$ are disjoint, and $Y = \bigcup_{\lambda \in \Lambda} \mathrm{im}(i_\lambda)$ is the (external) disjoint union of the $X_\lambda$.

A ''bouquet'' or ''wedge'' is a disjoint union of [[pointed set]]s.  It has the same universal property with respective to pointed maps.  There is a similar explicit construction.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  P. R. Halmos, ''Naive Set Theory'', Undergraduate Texts in Mathematics, Springer (1960) {{ISBN|0-387-90092-6}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  Tammo tom Dieck, ''Algebraic Topology'', European Mathematical Society (2008) {{ISBN|3037190485}}</TD></TR>
</table>