Editing Union (set theory) (section)

== Arbitrary union ==
The most general notion is the union of an arbitrary collection of sets, sometimes called an ''infinitary union''. If '''M''' is a set or [[Class (set theory)|class]] whose elements are sets, then ''x'' is an element of the union of '''M''' [[if and only if]] there is [[existential quantification|at least one]] element ''A'' of '''M''' such that ''x'' is an element of ''A''.<ref name=":1">{{Cite book |last=Smith |first=Douglas |url=https://archive.org/details/transitiontoadva0000smit |title=A Transition to Advanced Mathematics |last2=Eggen |first2=Maurice |last3=Andre |first3=Richard St |date=2014-08-01 |publisher=Cengage Learning |isbn=9781285463261 |language=en |url-access=registration}}</ref> In symbols:
: <math>x \in \bigcup \mathbf{M} \iff \exists A \in \mathbf{M},\ x \in A.</math>
This idea subsumes the preceding sections—for example, ''A'' ∪ ''B'' ∪ ''C'' is the union of the collection {{mset|''A'', ''B'', ''C''}}. Also, if '''M''' is the empty collection, then the union of '''M''' is the empty set.

=== Formal derivation ===
In [[Zermelo–Fraenkel set theory]] (ZFC) and other set theories, the ability to take the arbitrary union of any sets is granted by the [[axiom of union]], which states that, given any set of sets <math>A</math>, there exists a set <math>B</math>, whose elements are exactly those of the elements of <math>A</math>. Sometimes this axiom is less specific, where there exists a <math>B</math> which contains the elements of the elements of <math>A</math>, but may be larger. For example if <math>A = \{ \{1\}, \{2\} \},</math> then it may be that  <math>B = \{ 1, 2, 3\}</math> since <math>B</math> contains 1 and 2. This can be fixed by using the [[axiom of specification]] to get the subset of <math>B</math> whose elements are exactly those of the elements of <math>A</math>. Then one can use the [[axiom of extensionality]] to show that this set is unique. For readability, define the binary [[Predicate (logic)|predicate]] <math>\operatorname{Union}(X,Y)</math> meaning "<math>X</math> is the union of <math>
Y</math>" or "<math>X = \bigcup Y</math>" as:

<math display="block">\operatorname{Union}(X,Y) \iff \forall x (x \in X \iff \exists y \in Y ( x \in y))</math>

Then, one can prove the statement "for all <math>Y</math>, there is a unique <math>X</math>, such that <math>X</math> is the union of <math>
Y</math>":

<math display="block">\forall Y \, \exists ! X (\operatorname{Union}(X,Y))</math>

Then, one can use an [[extension by definition]] to add the union operator <math>\bigcup A</math> to the [[Zermelo–Fraenkel set theory#Formal language|language of ZFC]] as:

<math display="block">\begin{align}
B = \bigcup A   & \iff \operatorname{Union}(B,A) \\
                & \iff \forall x (x \in B \iff \exists y \in Y(x \in y))
\end{align}</math>

or equivalently:

<math display="block">x \in \bigcup A \iff \exists y \in A \, (x \in y)</math>

After the union operator has been defined, the binary union <math>A \cup B</math> can be defined by showing there exists a unique set <math>C = \{A,B\}</math> using the [[axiom of pairing]], and defining <math>A \cup B = \bigcup \{A,B\}</math>. Then, finite unions can be defined inductively as:

<math display="block">\bigcup _ {i=1} ^ 0 A_i = \varnothing \text{, and } \bigcup_{i=1}^n A_i = \left(\bigcup_{i=1}^{n-1} A_i \right) \cup A_n</math>