# Union (set theory)
In set theory, the **union** of two sets is the set whose members are exactly the objects that belong to at least one of them: for classes or sets A and B, the union A ∪ B is defined as A ∪ B = {x : (x ∈ A) ∨ (x ∈ B)}, so x belongs to A ∪ B if and only if x ∈ A or x ∈ B.[1](https://www.math.uwaterloo.ca/~randre/sets/Axiom_Book_Proofread.pdf) The same idea extends to a whole family of sets: if A is a non-empty class of classes, the union ⋃C∈A C contains exactly those x that belong to some C in A.[1](https://www.math.uwaterloo.ca/~randre/sets/Axiom_Book_Proofread.pdf)
| Key fact | Detail |
|---|---|
| Binary definition | A ∪ B = {x : (x ∈ A) ∨ (x ∈ B)}; membership requires belonging to at least one of the two sets[1](https://www.math.uwaterloo.ca/~randre/sets/Axiom_Book_Proofread.pdf) |
| General union | For a set of sets x, ⋃x contains exactly the elements of the members of x: z ∈ ⋃x ↔ ∃w ∈ x (z ∈ w)[2](https://math.ucla.edu/~marks/notes/set_theory_notes.pdf) |
| Axiomatic basis | The axiom of union asserts that for every set of sets x, the union ⋃x is itself a set[2](https://math.ucla.edu/~marks/notes/set_theory_notes.pdf) • [3](https://cs.lmu.edu/~ray/notes/settheory/) |
| Binary union from axioms | A ∪ B is ⋃{A, B}; the pair {A, B} exists by the axiom of pairing, and the axiom of union then applies to it[3](https://cs.lmu.edu/~ray/notes/settheory/) |
| Algebraic laws | Union is commutative (R ∪ S = S ∪ R), associative ((R ∪ S) ∪ T = R ∪ (S ∪ T)), and idempotent (R ∪ R = R)[4](https://www.onenoughtone.com/learn/set-operations/1) |
| Relation to logic | Union mirrors logical OR: x ∈ A ∪ B is the formula (x ∈ A) ∨ (x ∈ B)[1](https://www.math.uwaterloo.ca/~randre/sets/Axiom_Book_Proofread.pdf) • [5](https://web.sas.upenn.edu/ancoop/2018/10/10/set-operations-and-subsets/) |
| Use in databases | In relational algebra, union combines result tables, supporting tasks such as merging customer records from multiple regional databases and querying active and archived records together[4](https://www.onenoughtone.com/learn/set-operations/1) |
## Definition and notation
The binary union is defined by a membership criterion rather than by listing elements: x ∈ A ∪ B if and only if x ∈ A or x ∈ B.[1](https://www.math.uwaterloo.ca/~randre/sets/Axiom_Book_Proofread.pdf)
For a family of sets the notation is the large union: ⋃C∈A C = {x : x ∈ C for some C ∈ A}.[1](https://www.math.uwaterloo.ca/~randre/sets/Axiom_Book_Proofread.pdf) The complement operation, intersection, and union together form the basic class operations from which new classes and sets are constructed; Venn diagrams serve as a visual guide when interpreting them.[1](https://www.math.uwaterloo.ca/~randre/sets/Axiom_Book_Proofread.pdf)
## The axiom of union
In axiomatic set theory, union is not merely defined but guaranteed to exist. The [axiom of union](https://www.edgepedia.org/axiom-of-union) states that given any set of sets x, there is a set containing exactly all the elements of those sets, denoted ⋃x; precisely, y = ⋃x means ∀z[z ∈ y ↔ ∃w ∈ x (z ∈ w)].[2](https://math.ucla.edu/~marks/notes/set_theory_notes.pdf) The LMU formulation reads the same idea in first-order form: ∀A ∃U ∀x (x ∈ U ↔ ∃S ∈ A, x ∈ S), that is, given a set of sets, we can form the set containing all and only the elements of those sets.[3](https://cs.lmu.edu/~ray/notes/settheory/)
The binary union is then a special case. The set {A, B} exists by the [axiom of pairing](https://www.edgepedia.org/axiom-of-pairing), and since it is a set, the axiom of union applies to it, giving ⋃{A, B}; for convenience, when only two sets are involved, one writes A ∪ B for ⋃{A, B}.[3](https://cs.lmu.edu/~ray/notes/settheory/) In ZF set theory the same construction appears as axiom A5: A ∪ B, the union of the set {A, B}, exists.[6](https://www.math.wustl.edu/~freiwald/ch1.pdf) Texts differ in how they state the axiom: in one common formulation Union states that the union of a set is a set, while in another it simply states that ⋃x exists.[7](https://sci.mu.edu.iq/wp-content/uploads/2021/08/Foundations-of-Mathematicsmathetal.net_.pdf)
The axiom of union also combines with other axioms. Together with pairing it proves that x ∪ y exists for all sets x and y, since x ∪ y = ⋃{x, y}.[2](https://math.ucla.edu/~marks/notes/set_theory_notes.pdf) The axiom of infinity, which asserts an inductive set containing ∅ and closed under y ↦ y ∪ {y}, uses union in its very statement.[2](https://math.ucla.edu/~marks/notes/set_theory_notes.pdf)
## Algebraic laws
Union is an algebraic operation. It is commutative, associative, and idempotent: R ∪ S = S ∪ R, (R ∪ S) ∪ T = R ∪ (S ∪ T), and R ∪ R = R.[4](https://www.onenoughtone.com/learn/set-operations/1) These laws follow from propositional logic, since union is defined through disjunction: (x ∈ A) ∨ (x ∈ B) is equivalent to (x ∈ B) ∨ (x ∈ A), which gives commutativity directly from the definition.[5](https://web.sas.upenn.edu/ancoop/2018/10/10/set-operations-and-subsets/)
Union and intersection interact through distributive laws and [De Morgan's laws](https://www.edgepedia.org/de-morgans-laws), which are stated for classes and sets alongside the basic union and intersection operations.[1](https://www.math.uwaterloo.ca/~randre/sets/Axiom_Book_Proofread.pdf)
## Union in mathematics and applications
Union appears throughout mathematics as a way of building larger collections from smaller ones. In descriptive set theory, the Borel sets, the first hierarchy of definable sets introduced in the 1898 book of [Émile Borel](https://www.edgepedia.org/emile-borel), are obtained from the open sets by iterated application of the operations of countable union and complementation.[8](https://plato.stanford.edu/entries/settheory-early/) In constructive and intuitionistic variants of ZF, the union axiom likewise allows one, given a set X, to form a set Y whose elements are elements of elements of X.[9](https://plato.stanford.edu/archives/fall2024/entries/set-theory-constructive/axioms-CZF-IZF.html)
In computing, union is a core operation of relational algebra, the theoretical basis of database query languages. Practical applications include consolidating data, such as unifying customer records held in several regional databases; aggregating over time, for instance bringing together sales figures from separate months or quarters; queries spanning multiple sources that cover both live and archived records; breaking complex OR conditions into individual selections joined by a union; merging results from tables partitioned horizontally in distributed databases; and producing reports across several categories.[4](https://www.onenoughtone.com/learn/set-operations/1) Set-theoretic union types also appear in programming language type systems, where union, intersection, and negation type connectives support precise typing of branching, pattern matching, and function overloading.[10](https://arxiv.org/abs/2111.03354)
## References
1. [André, R. An Introduction to Axiomatic Reasoning, Chapter 3: Operations on classes and sets, University of Waterloo](https://www.math.uwaterloo.ca/~randre/sets/Axiom_Book_Proofread.pdf)
2. [Mark's Set Theory Notes, UCLA Department of Mathematics](https://math.ucla.edu/~marks/notes/set_theory_notes.pdf)
3. [Notes on Set Theory (axioms of pairing and union), Loyola Marymount University](https://cs.lmu.edu/~ray/notes/settheory/)
4. [Set Operations in Relational Algebra, OneNoughtOne](https://www.onenoughtone.com/learn/set-operations/1)
5. [Cooper, A. Set Operations and Subsets, University of Pennsylvania](https://web.sas.upenn.edu/ancoop/2018/10/10/set-operations-and-subsets/)
6. [Freiwald, J. Chapter 1: Sets, Washington University in St. Louis](https://www.math.wustl.edu/~freiwald/ch1.pdf)
7. [Foundations of Mathematics, Chapter 3: The Theory of Membership](https://sci.mu.edu.iq/wp-content/uploads/2021/08/Foundations-of-Mathematicsmathetal.net_.pdf)
8. [The Early Development of Set Theory, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/settheory-early/)
9. [Set Theory: Constructive and Intuitionistic ZF, Axioms of CZF and IZF, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/archives/fall2024/entries/set-theory-constructive/axioms-CZF-IZF.html)
10. [Castagna, G. Programming with union, intersection, and negation types, arXiv:2111.03354](https://arxiv.org/abs/2111.03354)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › Formal logic and foundations › Set theory › Elementary set theory*
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