Union of Sets
Calculus & Advanced MathThe 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. Duplicates are listed only once.
Example
$A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$. $A \cup B = \{1, 2, 3, 4, 5\}$. The $3$ appears in both but is listed only once in the union.
Key Insight
In a Venn diagram, the union is the entire shaded region covering both circles.
Definition
$A \cup B = \{x : x \in A \text{ or } x \in B\}$, where "or" is inclusive. Key properties: $A \cup A = A$ (idempotent), $A \cup \emptyset = A$ (identity), $A \cup B = B \cup A$ (commutative), $A \cup (B \cup C) = (A \cup B) \cup C$ (associative).
Example
$A = \{x \in \mathbb{Z} : x < 0\}$, $B = \{x \in \mathbb{Z} : x > 0\}$. $A \cup B$ = all nonzero integers. Note $0 \notin A \cup B$.
Key Insight
Union corresponds to logical OR. De Morgan's law connects union and intersection: $(A \cup B)' = A' \cap B'$.
Definition
The union of an arbitrary collection $\{A_i\}_{i \in I}$ is $\bigcup_{i \in I} A_i = \{x : x \in A_i \text{ for some } i \in I\}$. In measure theory, countable unions of measurable sets are measurable, but uncountable unions may not be. Sigma-algebras are defined by closure under countable unions.
Example
$\bigcup_{n=1}^{\infty} [1/n, 1] = (0, 1]$: the union approaches but never includes $0$. This illustrates that arbitrary unions of closed sets need not be closed.
Key Insight
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).