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.

Generalized Notation
Let $$S = S_1 \cup S_2 \cup \ldots \cup S_n$$. Then:

$$\bigcup_{i \in \mathbb{N}^*_n} S_i = \left\{{x: \exists i \in \mathbb{N}^*_n: x \in S_i}\right\}$$

If it is clear from the context that $$i \in \mathbb{N}^*_n$$, we can also write $$\bigcup_{\mathbb{N}^*_n} S_i$$.

An alternative notation for the same concept is $$\bigcup_{i=1}^n S_i$$.

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