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:

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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:

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

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
Canonical name	Union
Date of creation	2013-03-22 12:14:19
Last modified on	2013-03-22 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