Union

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[edit]

The union operation is:

• associative - (A ∪ B) ∪ C = A ∪ (B ∪ C)
• commutative - A ∪ B = B ∪ A.

General unions[edit]

Finite unions[edit]

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

${\displaystyle {\displaystyle \bigcup _{i=1}^n}{X}_i=X_1\cup (X_2\cup (X_3\cup (\cdots X_n)\cdots ))).}$

Infinite unions[edit]

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

${\displaystyle \bigcup _{\lambda \in \mathrm{\Lambda }}{X}_{\lambda }=\{x:\mathrm{\exists }\lambda \in \mathrm{\Lambda },x\in X_{\lambda }\}.}$

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:

${\displaystyle \bigcup X=\{x:\mathrm{\exists }Y\in X,x\in Y\}.}$

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

${\displaystyle A\cup B=\bigcup \{A,B\}.}$

See also[edit]

• Disjoint union
• Intersection

References[edit]

• 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.