Union of Sets
The union of two sets is the set of all elements that belong to either set (or both), written A ∪ B.
Formula
A \cup B = \{x : x \in A \text{ or } x \in B\}
Definition
The union of two sets combines everything in both sets into one big set; if an element appears in either set, it goes into the union, and duplicates are listed only once. Formally, $A \cup B = \{x : x \in A \text{ or } x \in B\}$, where "or" is inclusive, with properties $A \cup A = A$, $A \cup \emptyset = A$, commutativity, and associativity. For an arbitrary collection $\{A_i\}_{i \in I}$, the union is $\bigcup_{i \in I} A_i = \{x : x \in A_i \text{ for some } i \in I\}$; countable unions of measurable sets are measurable, but uncountable unions may not be, and sigma-algebras are defined by closure under countable unions.
Example
For $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$: $A \cup B = \{1, 2, 3, 4, 5\}$, the $3$ appears in both but is listed only once. For $A = \{x \in \mathbb{Z} : x < 0\}$ and $B = \{x \in \mathbb{Z} : x > 0\}$: $A \cup B$ is all nonzero integers. A subtler example, $\bigcup_{n=1}^{\infty} [1/n, 1] = (0, 1]$, shows that an infinite union approaches but never includes $0$, illustrating that arbitrary unions of closed sets need not be closed.
Key Insight
In a Venn diagram, the union is the entire shaded region covering both circles. Union corresponds to logical OR, and De Morgan's law connects union and intersection: $(A \cup B)' = A' \cap B'$. The distinction between finite, countable, and uncountable unions is central to measure theory and topology, where different closure conditions define different structures (rings, algebras, sigma-algebras).