Definition:Set Union

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

The union (or logical sum, or sum) of $$S$$ and $$T$$ is written $$S \cup T$$.

It means the set which consists of all the elements which are contained in either (or both) of $$S$$ and $$T$$:
 * $$x \in S \cup T \iff x \in S \or x \in T$$

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:


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

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

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

If $$\mathbb{S}$$ is a set of sets, then the union of $$\mathbb{S}$$ is:
 * $$\bigcup \mathbb{S} = \left\{{x: \exists S \in \mathbb{S}: x \in S}\right\}$$

That is, the set of all elements of all elements of $$\mathbb{S}$$.

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