Union (set theory)

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

This article uses mathematical symbols.

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 usually 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.

Finite unions

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 if x is in A or x is in B or x is in C.

Union is an associative operation, it doesn't matter in what order unions are taken. In mathematics a finite union means any union carried out on a finite number of sets: it doesn't imply that the union set is a finite set.

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 the sets can be written 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 distribute over each other, and all three operations are combined in De Morgan's laws. Replacing union with symmetric difference gives a Boolean ring instead of a Boolean algebra.

Infinite 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:

x \in \bigcup\mathbf{M} \iff \exists A{\in}\mathbf{M}, x \in A.

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 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 finite unions and logical disjunction extends to one between infinite unions and existential quantification.

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

\bigcup \mathbf{M},

while most people will instead write

\bigcup_{A\in\mathbf{M}} A.

The latter notation can be generalised to

\bigcup_{i\in I} A_{i},

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 to that of infinite series:

\bigcup_{i=1}^{\infty} A_{i}.

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 σ-algebras.) 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

\bigcup_{i\in I} (A \cap B_{i}) = A \cap \bigcup_{i\in I} B_{i}.

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

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

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Basic definition

Finite unions

Algebraic properties

Infinite unions

See also