Definition:Set Union

Definition
Let $S$ and $T$ be sets.

The (set) union of $S$ and $T$ is the set $S \cup T$, which consists of all the elements which are contained in either (or both) of $S$ and $T$:
 * $x \in S \cup T \iff x \in S \lor x \in T$

or, slightly more formally:
 * $A = S \cup T \iff \forall z: \left({z \in A \iff z \in S \lor z \in T}\right)$

We can write:
 * $S \cup T := \left\{{x: x \in S \lor x \in T}\right\}$

For example, let $S = \left \{{1, 2, 3}\right\}$ and $T = \left \{{2, 3, 4}\right\}$. Then $S \cup T = \left \{{1, 2, 3, 4}\right\}$.

It can be seen that, in this form, $\cup$ is a binary operation which acts on sets.

Axiomatic Set Theory
The concept of set union is axiomatised in the Axiom of Union in Zermelo-Fraenkel set theory:
 * $\forall A: \exists x: \forall y: \left({y \in x \iff \exists z: \left({z \in A \land y \in z}\right)}\right)$

Also known as
The union of sets is also known as the logical sum, or just sum, but these terms are usually considered old-fashioned nowadays.

The term join can also be seen, but this is usually reserved for specific contexts.

Some authors use the notation $S + T$ for $S \cup T$, but this is non-standard and can be confusing, so its use is not recommended.

Also, $S + T$ is sometimes used for disjoint union.

Also see

 * Definition:Set Intersection, a related operation.


 * Definition:Disjoint Union (Set Theory)


 * Union of Singleton, where it is shown that $\displaystyle \mathbb S = \left\{{S}\right\} \implies \bigcup \mathbb S = S$
 * Union of Empty Set, where it is shown that $\displaystyle \mathbb S = \varnothing \implies \bigcup \mathbb S = \varnothing$