Union (set theory)
In mathematics, 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. The union of A and B is standardly written "A ∪ B". Formally:
- x is an element of A ∪ B if and only if
- x is an element of A or
- x is an element of B.
(Note that in mathematics the word "or" allows both possibilities).
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 {2,3,5,7,11,...} and the set of even numbers {2,4,6,8,10,...}.
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)). Union is an associative operation; thus, A ∪ (B ∪ C ) = (A ∪ B) ∪ C.
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. In symbols:
- x ∈ ∪M ↔ ∃A∈M, x ∈ A.
This idea subsumes the above paragraphs, 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. 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 formal set theory.
The notation for this last concept can vary considerably. Hardcore set theorists will simply write "∪M", while most people will instead write "∪A∈M A". The latter notation can be generalised to "∪i∈I Ai", 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 I.
In the case that the index set I is the set of natural numbers, you might see notation analogous that that of summation:
- ∞
- ∪ 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 ⋃, but until then, try <big>∪</big>.)
See also: Basic set theory