Union

Idea

The union of a family of sets is the set of all elements belonging to at least one set in the family.

If $S:I\to \mathrm{Set}$ is an $I$-indexed family of sets, its union is written

$$\bigcup _{i\in I}{S}_i$$

and is defined by

$$x\in \bigcup _{i\in I}{S}_i⟺\exists i:I.x\in S_i.$$

Definition

Let $I:\mathrm{Set}$ and let $S:I\to \mathrm{Set}$. Then

$$\bigcup _{i\in I}S(i):=\{x\mid \exists i\in I.x\in S(i)\}.$$

If all the sets $S(i)$ are subsets of some ambient set $A$, this may also be written as

$$\bigcup _{i\in I}S(i)\subseteq A.$$

Set of sets form

If $\mathcal{S}$ is a set whose elements are themselves sets, then its union is the set

$$\bigcup \mathcal{S}=\{x\mid \exists U\in \mathcal{S}.x\in U\}.$$

Thus

$$x\in \bigcup \mathcal{S}⟺\exists U\in \mathcal{S}.x\in U.$$

Special cases

For two sets $A,B$, their union is

$$A\cup B=\{x\mid x\in A\vee x\in B\}.$$

More generally:

• a finite union is a union indexed by a finite set
• a countable union is a union indexed by a countable set
• an uncountable union is a union indexed by an uncountable set

Closure under unions

A collection of sets $\mathcal{C}$ is said to be closed under unions of a given kind if the union of any family of sets in $\mathcal{C}$, indexed by a set of that kind, again lies in $\mathcal{C}$.

For example, $\mathcal{C}$ is closed under finite unions if whenever

$$S:I\to \mathcal{C}$$

with $I$ finite, we have

$$\bigcup _{i\in I}S(i)\in \mathcal{C}.$$

Similarly:

• closed under countable unions means this holds for countable $I$
• closed under arbitrary unions means this holds for all index sets $I$

If one works inside an ambient set $A$, so that $\mathcal{C}\subseteq \mathcal{P}(A)$, this can be written as

$$\forall S:I\to \mathcal{C},\bigcup _{i\in I}S(i)\in \mathcal{C}.$$

Characterisation

If $\mathcal{S}$ is a set of subsets of $A$, then its union is the unique subset $T\subseteq A$ such that

$$\forall x\in A.x\in T⟺\exists U\in \mathcal{S}.x\in U.$$

Remarks

• The union operation is one of the basic constructions of set theory, and appears in the axioms of ZF as the axiom of union.
• In topology, open sets are required to be closed under arbitrary unions.