Union: Difference between revisions

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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.
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.


==Properties==
The union operation is:
The union operation is:
* [[associative]] - (A ∪ B) ∪ C = A ∪ (B ∪ C)
* [[associative]] - (A ∪ B) ∪ C = A ∪ (B ∪ C)
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==See also==
==See also==
* [[Disjoint union]]
* [[Disjoint union]]
* [[Intersection]]


==References==
==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 }}
* {{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 }}
* {{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 }}
* {{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 }}

Revision as of 15:33, 4 November 2008

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