Union (set theory)

Union of two sets:
<math>~A \cup B</math>

Union of three sets:
<math>~A \cup B \cup C</math>

File:Example of a non pairwise disjoint family of sets.svg

The union of A, B, C, D, and E is everything except the white area.

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> It is one of the fundamental operations through which sets can be combined and related to each other. A Template:Visible anchor refers to a union of [[Zero|zero (Template:Tmath)]] 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.

Binary unionEdit

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.<ref name=":3">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In set-builder notation,

<math>A \cup B = \{ x: x \in A \text{ or } x \in B\}</math>.<ref name=":0">Template:Cite book</ref>

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:

A = Template:Mset

B = Template:Mset

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

Sets cannot have duplicate elements,<ref name=":0" /><ref>Template:Cite book</ref> so the union of the sets Template:Mset and Template:Mset is Template:Mset. Multiple occurrences of identical elements have no effect on the cardinality of a set or its contents.

Finite unionsEdit

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.<ref>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

NotationEdit

The notation for the general concept can vary considerably. For a finite union of sets <math>S_1, S_2, S_3, \dots , S_n</math> one often writes <math>S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n</math> or <math display="inline">\bigcup_{i=1}^n S_i</math>. Various common notations for arbitrary unions include <math display="inline">\bigcup \mathbf{M}</math>, <math display="inline">\bigcup_{A\in\mathbf{M}} A</math>, and <math display="inline">\bigcup_{i\in I} A_{i}</math>. The last of these notations refers to the union of the collection <math>\left\{A_i : i \in I\right\}</math>, where I is an index set and <math>A_i</math> is a set for every Template:Tmath. In the case that the index set I is the set of natural numbers, one uses the notation <math display="inline">\bigcup_{i=1}^{\infty} A_{i}</math>, which is analogous to that of the infinite sums in series.<ref name=":1" />

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

Notation encodingEdit

In Unicode, union is represented by the character Template:Unichar.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In TeX, <math>\cup</math> is rendered from \cup and <math display="inline">\bigcup</math> is rendered from \bigcup.

Arbitrary unionEdit

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.<ref name=":1">Template:Cite book</ref> In symbols:

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

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

Formal derivationEdit

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, which states that, given any set of sets <math>A</math>, there exists a set <math>B</math>, whose elements are exactly those of the elements of <math>A</math>. Sometimes this axiom is less specific, where there exists a <math>B</math> which contains the elements of the elements of <math>A</math>, but may be larger. For example if <math>A = \{ \{1\}, \{2\} \},</math> then it may be that <math>B = \{ 1, 2, 3\}</math> since <math>B</math> contains 1 and 2. This can be fixed by using the axiom of specification to get the subset of <math>B</math> whose elements are exactly those of the elements of <math>A</math>. Then one can use the axiom of extensionality to show that this set is unique. For readability, define the binary predicate <math>\operatorname{Union}(X,Y)</math> meaning "<math>X</math> is the union of <math> Y</math>" or "<math>X = \bigcup Y</math>" as:

<math display="block">\operatorname{Union}(X,Y) \iff \forall x (x \in X \iff \exists y \in Y ( x \in y))</math>

Then, one can prove the statement "for all <math>Y</math>, there is a unique <math>X</math>, such that <math>X</math> is the union of <math> Y</math>":

<math display="block">\forall Y \, \exists ! X (\operatorname{Union}(X,Y))</math>

Then, one can use an extension by definition to add the union operator <math>\bigcup A</math> to the language of ZFC as:

<math display="block">\begin{align} B = \bigcup A & \iff \operatorname{Union}(B,A) \\

               & \iff \forall x (x \in B \iff \exists y \in Y(x \in y))

\end{align}</math>

or equivalently:

<math display="block">x \in \bigcup A \iff \exists y \in A \, (x \in y)</math>

After the union operator has been defined, the binary union <math>A \cup B</math> can be defined by showing there exists a unique set <math>C = \{A,B\}</math> using the axiom of pairing, and defining <math>A \cup B = \bigcup \{A,B\}</math>. Then, finite unions can be defined inductively as:

<math display="block">\bigcup _ {i=1} ^ 0 A_i = \varnothing \text{, and } \bigcup_{i=1}^n A_i = \left(\bigcup_{i=1}^{n-1} A_i \right) \cup A_n</math>

Algebraic propertiesEdit

Template:See also

Binary union is an associative operation; that is, for any sets Template:Tmath, <math display="block">A \cup (B \cup C) = (A \cup B) \cup C.</math> Thus, the parentheses may be omitted without ambiguity: either of the above can be written as Template:Tmath. Also, union is commutative, so the sets can be written in any order.<ref>Template:Cite book</ref> The empty set is an identity element for the operation of union. That is, Template:Tmath, for any set Template:Tmath. Also, the union operation is idempotent: Template:Tmath. All these properties follow from analogous facts about logical disjunction.

Intersection distributes over union <math display="block">A \cap (B \cup C) = (A \cap B)\cup(A \cap C)</math> and union distributes over intersection<ref name=":3" /> <math display="block">A \cup (B \cap C) = (A \cup B) \cap (A \cup C).</math> The power set of a set Template:Tmath, 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 <math display="block">A \cup B = ( A^\complement \cap B^\complement )^\complement,</math> where the superscript <math>{}^\complement</math> denotes the complement in the universal set Template:Tmath. Alternatively, intersection can be expressed in terms of union and complementation in a similar way: <math>A \cap B = ( A^\complement \cup B^\complement )^\complement</math>. These two expressions together are called De Morgan's laws.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

History and etymologyEdit

Template:Further The english word union comes from the term in middle French meaning "coming together", which comes from the post-classical Latin unionem, "oneness".<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The original term for union in set theory was Vereinigung (in german), which was introduced in 1895 by Georg Cantor.<ref>Template:Cite journal</ref> The english use of union of two sets in mathematics began to be used by at least 1912, used by James Pierpont.<ref>Template:Cite book</ref><ref>Oxford English Dictionary, “union (n.2), sense III.17,” March 2025, https://doi.org/10.1093/OED/1665274057</ref> The symbol <math>\cup</math> used for union in mathematics was introduced by Giuseppe Peano in his Arithmetices principia in 1889, along with the notations for intersection <math>\cap</math>, set membership <math>\in</math>, and subsets <math>\subset</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

NotesEdit

Template:Reflist

External linksEdit

Template:Springer
Infinite Union and Intersection at ProvenMath De Morgan's laws formally proven from the axioms of set theory.

Template:Set theory Template:Mathematical logic