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.

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

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


 * $$\bigcup_{i=1}^n X_i = X_1 \cup (X_2 \cup (X_3 \cup (\cdots X_n)\cdots))) . \, $$

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


 * $$\bigcup_{\lambda\in \Lambda} X_\lambda = \{ x : \exists \lambda \in \Lambda,~x \in X_\lambda \} .\, $$

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:


 * $$\bigcup X = \{ x : \exists Y \in X,~ x \in Y \} . \,$$

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


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