Combine no sets at all and set theory says you get the empty set
The union of sets gathers every element that appears in any of them. Merge {1, 2, 3} with {2, 3, 4} and you get {1, 2, 3, 4}, since a set never lists anything twice. Push the idea to its limit and a union of zero sets is, by definition, empty: nothing collected yields nothing.
Union is one of the basic ways of building new sets from old ones, written with the cup-shaped symbol ∪. For two sets, it contains whatever lies in the first, the second, or both. Take {1, 3, 5, 7} and {1, 2, 4, 6, 7}: their union is the seven numbers from 1 to 7. A negative example helps too. The number 9 is missing from the union of the primes and the even numbers, since it is neither prime nor even.
Nothing restricts the count to two. The union of three sets holds everything found in at least one of them and nothing else, and the same logic stretches to any finite number. A finite union means only that finitely many sets are being combined; the result can still be infinite, as when the sets themselves are endless collections of integers.
The most general version handles an arbitrary collection, even an infinite one. An object belongs to the union of a collection exactly when it sits inside at least one member of that collection. This covers every earlier case, since combining three sets is just the union of the collection containing those three, and it explains the empty result: if the collection has no members, nothing can qualify.
In Zermelo–Fraenkel set theory, the standard foundation of mathematics, the right to form such unions is not assumed but granted by a dedicated rule, the axiom of union. Paired with other axioms, it guarantees that the union of any set of sets exists and is unique. Notation varies, and when the symbol precedes a list rather than sitting between two sets it is usually drawn larger, much like the sigma used for infinite sums.
Source: Union (set theory)
