Union (set theory)

In set theory and other branches of mathematics, the union of some sets is the set that contains everything that belongs to any of the sets, but nothing else.

Basic definition

If A and B are sets, then the union of 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.

(This is an inclusive "or".)

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,...}, because 9 is neither prime nor even.

More generally, one can take the union of several sets at once. The union of A, B, and C, for example, contains all elements of A, all elements of B, and all elements of C, and nothing else. Formally, x is an element of A ∪ B ∪ C ∪ D iff x is in A or x is in B or x is in C.

Algebraic properties

Binary union (the union of just two sets at a time) is an associative operation; that is, A ∪ (B ∪ C) = (A ∪ B) ∪ C. In fact, A ∪ B ∪ C is equal to both of these sets as well, so parentheses are never needed when writing only unions. Similarly, union is commutative, so you can write the sets in any order. The empty set is an identity element for the operation of union. That is, {} ∪ A = A, for any set A. Thus one can think of the empty set as the union of zero sets. In terms of the definitions, these facts follow from analogous facts about logical disjunction.

Together with intersection and complement, union makes any power set into a Boolean algebra. For example union and intersection distributes over each other, and all three operations are combined in de Morgan's laws. If you want a Boolean ring instead of a Boolean algebra, then you can replace union with symmetric difference.

Infinitary unions

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:

<math>x \in \bigcup\mathbf{M} \iff \exists A{\in}\mathbf{M}, x \in A.</math>

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.

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. The analogy between finitary unions and logical disjunction extends to one between infinitary unions and existential quantification.

The notation for the general concept can vary considerably. Hardcore set theorists will simply write

<math>\bigcup \mathbf{M},</math>

while most people will instead write

<math>\bigcup_{A\in\mathbf{M}} A.</math>

The latter notation can be generalised to

<math>\bigcup_{i\in I} A_{i},</math>

which refers to the union of the collection {A_i : i is in I}. Here I is a set, and A_i is a set for every i in I. In the case that the index set[?] I is the set of natural numbers, the notation is analogous that that of summation:

<math>\bigcup_{i=1}^{\infty} A_{i}.</math>

When formatting is difficult, this can also be written "A₁ ∪ A₂ ∪ A₃ ∪ ···". (This last example, a union of countably many sets, is very common in analysis; 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 is of a larger size.

Intersection distributes over infinitary union, in the sense that

<math>\bigcup_{i\in I} (A \cap B_{i}) = A \cap \bigcup_{i\in I} B_{i}.</math>

We can also combine ifinitary union with infinitary intersection to get the law

<math>\bigcup_{i\in I} (\bigcap_{j\in J} A_{i,j}) \subseteq \bigcap_{j\in J} (\bigcup_{i\in I} A_{i,j}).</math>