Union (set theory)
Set operation combining all elements from given sets.
In set theory, the union of a collection of sets (symbolized by ∪) is the set made up of every element that appears in any of the sets. It is a basic operation for combining sets. A union of zero sets—called a nullary union—is defined to be the empty set.
For two sets, A and B, their union A ∪ B contains all elements that are in A, in B, or in both. 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}. With infinite sets, such as A = {even integers greater than 1} and B = {odd integers greater than 1}, the union is {2, 3, 4, 5, 6, …}. The number 9 is not in the union of the set of prime numbers and the set of even numbers, because 9 is neither prime nor even. Since sets cannot have duplicates, the union of {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}.
A union can involve more than two sets. For instance, the union of three sets A, B, and C contains every element that belongs to at least one of them. A finite union refers to the union of a finite number of sets, though the resulting set itself may be infinite.
Notation varies. For a finite collection S₁, S₂, S₃, …, Sₙ, the union is often written S₁ ∪ S₂ ∪ S₃ ∪ … ∪ Sₙ or ⋃_{i=1}^{n} Sᵢ. For arbitrary collections, common notations include ⋃ M, ⋃_{A∈M} A, and ⋃_{i∈I} Aᵢ, where I is an index set and each Aᵢ is a set. If I is the set of natural numbers, the notation ⋃_{i=1}^{∞} Aᵢ is used, similar to infinite series. When the union symbol appears before other symbols, it is usually drawn larger.
In Unicode, the union symbol is U+222A ∪. In TeX, \cup gives ∪ and \bigcup gives ⋃. In Typst, union gives ∪ and union.big gives ⋃.
The most general case 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 belongs to the union of M if and only if there exists at least one set A in M such that x belongs to A.
- field
- Set theory
- known_for
- Union of sets, nullary union, arbitrary union
- notation
- ∪ (binary), ⋃ (big union)
- axiom
- Axiom of union in Zermelo–Fraenkel set theory
Lore & Background
The union of two sets A and B is the set of elements which are in A, in B, or in both. In set-builder notation, A ∪ B = {x : x ∈ A or x ∈ B}. 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}. Sets cannot have duplicate elements, so the union of {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}.
A nullary union refers to a union of zero sets and is by definition equal to the empty set. Finite unions can be taken 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. 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.
Reader's Guide
The union operation is central to set theory and its applications across mathematics. It allows the combination of multiple sets into one, forming the basis for more complex set constructions. The concept extends from binary unions to finite unions and ultimately to arbitrary unions of any collection of sets, including infinite collections. The nullary union (union of zero sets) is defined as the empty set, providing a consistent base case. In Zermelo–Fraenkel set theory, the ability to take arbitrary unions is granted by the axiom of union, which states that given any set of sets A, there exists a set B whose elements are exactly those of the elements of A. This axiom, combined with the axiom of specification and extensionality, ensures the existence and uniqueness of the union. The notation varies: for finite unions one writes S₁ ∪ S₂ ∪ ... ∪ Sₙ or ⋃_{i=1}^{n} S_i; for arbitrary unions, notations include ⋃M, ⋃_{A∈M} A, and ⋃_{i∈I} A_i. In Unicode, union is represented by U+222A ∪ UNION; in TeX, \cup renders ∪ and \bigcup renders ⋃.
Did You Know?
- A nullary union (union of zero sets) is by definition equal to the empty set.
- The union of the set of prime numbers and the set of even numbers does not contain 9, because 9 is neither prime nor even.
- In Zermelo–Fraenkel set theory, the ability to take arbitrary unions is granted by the axiom of union.
- The union of two sets A and B is the set of elements which are in A, in B, or in both.
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