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)
Famous quotes containing the words arbitrary and/or unions:
“We do the same thing to parents that we do to children. We insist that they are some kind of categorical abstraction because they produced a child. They were people before that, and they’re still people in all other areas of their lives. But when it comes to the state of parenthood they are abruptly heir to a whole collection of virtues and feelings that are assigned to them with a fine arbitrary disregard for individuality.”
—Leontine Young (20th century)
“When Hitler attacked the Jews ... I was not a Jew, therefore, I was not concerned. And when Hitler attacked the Catholics, I was not a Catholic, and therefore, I was not concerned. And when Hitler attacked the unions and the industrialists, I was not a member of the unions and I was not concerned. Then, Hitler attacked me and the Protestant church—and there was nobody left to be concerned.”
—Martin Niemller (1892–1984)