Union (set theory)
In set theory, union (denoted as ∪) is a set operation between two sets that forms a set containing the elements of both sets.
Formally, the union A ∪ B is defined by the following: a ∈ A ∪ B if and only if ( a ∈ A ) ∨ ( a ∈ B ), where ∨ - is logical or. We see this connection between ∪ and ∨ symbols.
The union operation is:
- associative - (A ∪ B) ∪ C = A ∪ (B ∪ C)
- commutative - A ∪ B = B ∪ A.
The union of any finite number of sets may be defined inductively, as
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
- Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold. Section 4.
- Keith J. Devlin (1979). Fundamentals of Contemporary Set Theory. Springer-Verlag, 5,10. ISBN 0-387-90441-7.
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