Union of sets

sum of sets

One of the basic operations on (collections of) sets. Suppose one has some (finite or infinite) collection $\mathcal{K}$ of sets. Then the collection of all elements that belong to at least one of the sets in $\mathcal{K}$ is called the union, or, more rarely, the sum, of (the sets in) $\mathcal{K}$; it is denoted by $\bigcup \mathcal{K}$.

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In case $\mathcal{K}=\{A_{\alpha }:\alpha \in I\}$, the union is also denoted by $\bigcup _{\alpha }{A}_{\alpha }$, $\bigcup _{\alpha \in I}{A}_{\alpha }$, $\bigcup _{A\in \mathcal{K}}A$, or, more rarely, by $\sum _{\alpha }{A}_{\alpha }$.

In the Zermelo–Fraenkel axiom system for set theory, the sum-set axiom expresses that the union of a set of sets is a set.

If the sets $A_{\alpha }$ are disjoint, then in the category $\mathbf{Set}$ the union of the objects $A_{\alpha }$ is the sum of these objects in the categorical sense. In general, the sum of objects $X_{\alpha }$ is the disjoint union $\coprod _{\alpha }{X}_{\alpha }=\{(x,\alpha ):x\in X_{\alpha }\}$. The natural imbeddings $i_{\alpha }:X_{\alpha }\to \coprod _{\alpha }{X}_{\alpha }$ are given by $i_{\alpha }(x)=(x,\alpha )$. Thus, $\coprod _{\alpha }{X}_{\alpha }$ together with the $i_{\alpha }$, $\alpha \in I$, satisfies the universal property for categorical sums: For every family of mappings $f_{\alpha }:X_{\alpha }\to Y$ there is a unique mapping $f:\coprod _{\alpha }{X}_{\alpha }\to Y$ such that $f{i}_{\alpha }=f_{\alpha }$.

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