Definition:Set Union

Let $$S$$ and $$T$$ be any two sets.

The union (or sum) of $$S$$ and $$T$$ is written $$S \cup T$$ and means the set which consists of all the elements which are contained in either (or both) of $$S$$ and $$T$$.

$$x \in S \cup T \iff x \in S \lor x \in T$$

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 $$\cup$$ is an operator.

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.

Axiomatic Set Theory
The concept of set union is axiomatised in the Axiom of Unions in Zermelo-Fraenkel set theory.