union
The union of two sets $A$ and $B$ is the set which contains all $x\in A$ and all $x\in B$, denoted $A\cup B$. In the Venn diagram^{} below, $A\cup B$ is the entire area shaded in blue.
We can extend this to any (finite or infinite^{}) family ${({A}_{i})}_{i\in I}$, writing ${\bigcup}_{i\in I}{A}_{i}$ for the union of this family. Formally, for a family ${({A}_{i})}_{i\in I}$ of sets:
$$x\in \bigcup _{i\in I}{A}_{i}\iff \underset{i\in I}{\bigvee}(x\in {A}_{i})$$ 
Alternatively, and equivalently,
$$x\in \bigcup _{i\in I}{A}_{i}\iff \exists i\in I\text{such that}x\in {A}_{i}$$ 
This characterization^{} makes it much clearer that if $I$ is itself the empty set^{} (that is, if we are taking the union of an empty family), then the union is empty; that is,
$$\bigcup _{i\in \mathrm{\varnothing}}{A}_{i}=\mathrm{\varnothing}$$ 
Often elements of sets are taken from some universe^{} $U$ of elements under consideration (for example, the real numbers $\mathbb{R}$, or living things on the planet, or words in a particular book). When this is the case, it is meaningful to discuss the complement of a set: if $A$ is a set of elements from some universe $U$, then the complement of $A$ is the set
$${A}^{C}=U\backslash A=\{x\in U\mid x\notin A\}$$ 
From an axiomatic point of view, the existence of the union is guaranteed by the axiom of union.
Note that the sets ${A}_{i}$ may be, but need not be, disjoint. Unions satisfy some basic properties that are obvious from the definitions:

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Idempotency: $A\cup A=A$

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$A\cup {A}^{C}=U$ where $U$ is the universe of $A$

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Commutativity: $A\cup B=B\cup A$

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Associativity: $(A\cup B)\cup C=A\cup (B\cup C)$
Here are some examples of set unions:
$$\mathrm{\{}1,2\}\cup \{3,4\}=\{1,2,3,4\}$$  
$$\mathrm{\{}blue,green\}\cup \mathrm{\varnothing}=\{blue,green\}$$  
$$\mathrm{\{}x\in \mathbb{Z}\mid x\u2a7e1\}\cup \{x\in \mathbb{Z}\mid x\u2a7d1\}=\{x\in \mathbb{Z}\mid x\ne 0\}$$  
$$\mathrm{\{}1,2\}\cup \{1,4\}=\{1,2,4\}$$  
$$\mathrm{\{}x\in \mathbb{R}\mid x\u2a7e2\}\cup \{x\in \mathbb{R}\mid x\u2a7d2\}=\mathbb{R}$$  
$$\bigcup _{\begin{array}{c}n\in \mathbb{Z}\\ n>0\end{array}}\{\frac{p}{n}\mid p\in \mathbb{Z}\}=\mathbb{Q}$$ 
The first three of these are the union of disjoint sets, while the latter three are not  in those cases, the sets overlap each other.
Title  union 

Canonical name  Union 
Date of creation  20130322 12:14:19 
Last modified on  20130322 12:14:19 
Owner  rm50 (10146) 
Last modified by  rm50 (10146) 
Numerical id  13 
Author  rm50 (10146) 
Entry type  Definition 
Classification  msc 03E30 
Related topic  Intersection^{} 