# 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:

 $x\in\bigcup_{i\in I}A_{i}\;\Leftrightarrow\;\bigvee_{i\in I}\,(x\in A_{i})$

Alternatively, and equivalently,

 $x\in\bigcup_{i\in I}A_{i}\;\Leftrightarrow\;\exists i\in I\text{ such that }x% \in A_{i}$

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,

 $\bigcup_{i\in\emptyset}A_{i}=\emptyset$

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:

• Idempotency: $A\cup A=A$

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

• Commutativity: $A\cup B=B\cup A$

• Associativity: $(A\cup B)\cup C=A\cup(B\cup C)$

Here are some examples of set unions:

 $\displaystyle\{1,2\}\cup\{3,4\}=\{1,2,3,4\}$ $\displaystyle\{blue,green\}\cup\emptyset=\{blue,green\}$ $\displaystyle\{x\in\mathbb{Z}\ \mid\ x\geqslant 1\}\cup\{x\in\mathbb{Z}\ \mid% \ x\leqslant-1\}=\{x\in\mathbb{Z}\ \mid\ x\neq 0\}$ $\displaystyle\{1,2\}\cup\{1,4\}=\{1,2,4\}$ $\displaystyle\{x\in\mathbb{R}\ \mid\ x\geqslant 2\}\cup\{x\in\mathbb{R}\ \mid% \ x\leqslant 2\}=\mathbb{R}$ $\displaystyle\bigcup_{\begin{subarray}{c}n\in\mathbb{Z}\\ n>0\end{subarray}}\left\{\frac{p}{n}\,\mid\,p\in\mathbb{Z}\right\}=\mathbb{Q}$

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 Union 2013-03-22 12:14:19 2013-03-22 12:14:19 rm50 (10146) rm50 (10146) 13 rm50 (10146) Definition msc 03E30 Intersection