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In [[set theory]], union (denoted as ∪) is a set operation between two sets that forms a set containing the elements of both sets.
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In [[set theory]], '''union''' (denoted as ∪) is a set operation between two sets that forms a set containing the elements of both sets.


Formally, union A ∪ B means that if a ∈ A ∪ B, then a ∈ A ∨ a ∈ B, where ∨ - is logical or. We see this connection between ∪ and ∨ symbols.
Formally, union A ∪ B means that if a ∈ A ∪ B, then a ∈ A ∨ a ∈ B, where ∨ - is logical or. We see this connection between ∪ and ∨ symbols.


Union operation is:
==Properties==
The union operation is:
* [[associative]] - (A ∪ B) ∪ C = A ∪ (B ∪ C)
* [[associative]] - (A ∪ B) ∪ C = A ∪ (B ∪ C)
* [[commutative]] - A ∪ B = B ∪ A.
* [[commutative]] - A ∪ B = B ∪ A.


==Forms==
==General unions==
===Finite unions===
===Finite unions===
The union of any finite number of sets may be defined inductively, as
:<math>\bigcup_{i=1}^n X_i = X_1 \cup (X_2 \cup (X_3 \cup (\cdots X_n)\cdots))) . \, </math>
===Infinite unions===
===Infinite unions===
The union of a general family of sets ''X''<sub>λ</sub> as λ ranges over a general index set Λ may be written in similar notation as
:<math>\bigcup_{\lambda\in \Lambda} X_\lambda = \{ x : \exists \lambda \in \Lambda,~x \in X_\lambda \} .\, </math>
We may drop the indexing notation and define the union of a set to be the set of elements of the elements of that set:
:<math>\bigcup X = \{ x : \exists Y \in X,~ x \in Y \} . \,</math>
In this notation the union of two sets ''A'' and ''B'' may be expressed as
:<math>A \cup B = \bigcup \{ A, B \} . \, </math>


==See also==
==See also==
* [[Disjoint union]]
* [[Disjoint union]]
* [[Intersection]]
==References==
* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[Van Nostrand Reinhold]] | year=1960 }}  Section 4.
* {{cite book | author=Keith J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90441-7 | pages=5,10 }}

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In set theory, union (denoted as ∪) is a set operation between two sets that forms a set containing the elements of both sets.

Formally, union A ∪ B means that if a ∈ A ∪ B, then a ∈ A ∨ a ∈ B, where ∨ - is logical or. We see this connection between ∪ and ∨ symbols.

Properties

The union operation is:

General unions

Finite unions

The union of any finite number of sets may be defined inductively, as

Infinite unions

The union of a general family of sets Xλ as λ ranges over a general index set Λ may be written in similar notation as

We may drop the indexing notation and define the union of a set to be the set of elements of the elements of that set:

In this notation the union of two sets A and B may be expressed as

See also

References