Union (set theory)
The union of two sets A and B is the set that contains all elements of A and all elements of B, but no other elements. Formally: x is element of the union if and only if x is an element of A or x is an element of B. The union of A and B is written "A ∪ B".
For example, the union of the sets {1,2,3} and {2,3,4} is {1,2,3,4}. The number 9 is not contained in the union of the set of prime numbers and the set of even numbers.
More generally, one can take the union of several sets at once. The union of A, B, C, and D, for example, is A ∪ B ∪ C ∪ D = A ∪ (B ∪ (C ∪ D)). (See Associative.)
The most general notion is the union of an arbitrary collection of sets. If M is a set whose elements are themselves sets, then x is an element of the union of M if and only if for at least one element A of M, x is an element of A. This subsumes the above paragraphs, in that for example, A ∪ B ∪ C is the union of the collection {A,B,C}.
The notation for this last concept can vary considerably. Hardcore set theorists will simply write "∪M", while most people will instead write "∪A∈MA". The latter notation can be generalised to "∪i∈IAi", which refers to the union of the collection {Ai : i ∈ I}. Here I is a set, and Ai is a set for every i in A. In the case that the index set I is the set of natural numbers, you might see notation like
- ∞
- ∪ Ai
- i=1
When formatting is difficult (as on these web pages), this can also be written "A1 ∪ A2 ∪ A3 ∪ ...", even though strictly speaking, A1 ∪ (A2 ∪ (A3 ∪ ... makes no sense. (This last example, a union of countably many sets, is actually very common; for an example see the article on σalgebras.) Finally, let us note that whenever the symbol "∪" is placed before other symbols instead of between them, it should be of a larger size (eventually this will be available in HTML as the character entity ⋃).
See also: