Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

[1] It is one of the fundamental operations through which sets can be combined and related to each other.

⁠) sets and it is by definition equal to the empty set.

For explanation of the symbols used in this article, refer to the table of mathematical symbols.

A more elaborate example (involving two infinite sets) is: As another example, 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.

Multiple occurrences of identical elements have no effect on the cardinality of a set or its contents.

Binary union is an associative operation; that is, for any sets ⁠

Thus, the parentheses may be omitted without ambiguity: either of the above can be written as ⁠

Also, union is commutative, so the sets can be written in any order.

[5] The empty set is an identity element for the operation of union.

Also, the union operation is idempotent: ⁠

All these properties follow from analogous facts about logical disjunction.

Intersection distributes over union

and union distributes over intersection[2]

⁠, together with the operations given by union, intersection, and complementation, is a Boolean algebra.

In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula

denotes the complement in the universal set ⁠

Alternatively, intersection can be expressed in terms of union and complementation in a similar way:

These two expressions together are called De Morgan's laws.

[6][7][8] One can take the union of several sets simultaneously.

Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set.

[9][10] The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union.

[11] In symbols: This idea subsumes the preceding sections—for example, A ∪ B ∪ C is the union of the collection {A, B, C}.

The notation for the general concept can vary considerably.

For a finite union of sets

Various common notations for arbitrary unions include

The last of these notations refers to the union of the collection

In the case that the index set I is the set of natural numbers, one uses the notation

, which is analogous to that of the infinite sums in series.

[11] When the symbol "∪" is placed before other symbols (instead of between them), it is usually rendered as a larger size.

Union of two sets:
Union of three sets:
The union of A, B, C, D, and E is everything except the white area.