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Wikipedia / Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elementss in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of zero (0) 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.

Union of two sets

The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. In set-builder notation,

  1. .

For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6, 7} then A ∪ B = {1, 2, 3, 4, 5, 6, 7}. A more elaborate example (involving two infinite sets) is:

  1. A =

  2. B =

As another example, the number 9 is not contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even.

Sets cannot have duplicate elements, so the union of the sets and is .

Finite unions

One can take the union of several sets simultaneously. For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. 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.

Notation

, where I is an index set is a set for every . In the case that the index set I is the set of natural numbers, which is analogous to that of the infinite sums in series.

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

Notation encoding

In Unicode, union is represented by the character . In TeX is rendered from \cup is rendered from \bigcup; In Typst, union, where union.big.

Arbitrary union

The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. If M is a set or class whose elements are sets, then x is an element of the union of M if and only if there is at least one element A of M such that x is an element of A. In symbols:

This idea subsumes the preceding sections—for example, A ∪ B ∪ C is the union of the collection . Also, if M is the empty collection, then the union of M is the empty set.

Formal derivation

In Zermelo–Fraenkel set theory (ZFC) and other set theories, the ability to take the arbitrary union of any sets is granted by the axiom of union contains 1 and 2. This can be fixed by using the axiom of specification. Then one can use the axiom of extensionality to show that this set is unique. For readability, define the binary predicate" as:

":

Then, one can use an extension by definition to the language of ZFC as:

or equivalently:

using the axiom of pairing. Then, finite unions can be defined inductively as:

Algebraic properties

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. The empty set is an identity element for the operation of union. That is, , for any set . Also, the union operation is idempotent: . All these properties follow from analogous facts about logical disjunction.

and union distributes over intersection The power set of a set , together with the operations given by union, intersection, and complementation, is a Boolean algebra denotes the complement in the universal set . These two expressions together are called De Morgan's laws.

History and etymology

The english word union comes from the term in middle French meaning "coming together", which comes from the post-classical Latin unionem, "oneness". The original term for union in set theory was Vereinigung (in german), which was introduced in 1895 by Georg Cantor. The english use of union of two sets in mathematics began to be used by at least 1912, used by James Pierpont. used for union in mathematics was introduced by Giuseppe Peano in his Arithmetices principia.

See also

  • − the union of sets of strings

Notes