Definition:Set Union

Definition
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 $$T$$:
 * $$x \in S \cup T \iff x \in S \or x \in T$$

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

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

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 $$\cup$$ 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:
 * $$\bigcup \mathbb S = \left\{{x: \exists X \in \mathbb S: x \in X}\right\}$$

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

Thus:
 * $$S \cup T = \bigcup \left\{{S, T}\right\}$$

Illustration by Venn Diagram
The red area in the following Venn diagram illustrates $$S \cup T$$:


 * VennDiagramSetUnion.png

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

Also see

 * Set Intersection, a related operation.


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