Arbitrary Unions
The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. If M is a set whose elements are themselves sets, then x is an element of the union of M if and only if there is at least one element A of M such that x is an element of A. In symbols:
That this union of M is a set no matter how large a set M itself might be, is the content of the axiom of union in axiomatic set theory.
This idea subsumes the preceding sections, in that (for example) A ∪ B ∪ C is the union of the collection {A,B,C}. Also, if M is the empty collection, then the union of M is the empty set. The analogy between finite unions and logical disjunction extends to one between arbitrary unions and existential quantification.
Read more about this topic: Union (set Theory)
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