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 $\bigcup\mathcal{K}$.

Comments

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 $\coprod_\alpha X_\alpha =\{(x,a):x\in X_\alpha\}$. 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 $fi_\alpha = f_\alpha$.

References

[1]	K. Kuratowski, "Introduction to set theory and topology" , Pergamon (1961) pp. 25 (Translated from French)