PlanetMath: union

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union

(Definition)

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.

$\begin{pspicture*}(0,0)(6,4) \pscircle[fillstyle=vlines,hatchcolor=blue,hatchwid... ...rput(1,2){$A$} \rput(5,2){$B$} \rput(-1,0){$.$} \rput(7,4){$.$} \end{pspicture*}$

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:

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

Alternatively, and equivalently,

$\displaystyle x \in \bigcup_{i\in I}A_i\; \Leftrightarrow \;\exists i\in I$ such that $\displaystyle x\in A_$

This characterization makes it much clearer that if

is itself the empty set (that is, if we are taking the union of an empty family), then the union is empty; that is,

$\displaystyle \bigcup_{i\in\emptyset}A_i=\emptyse$

Often elements of sets are taken from some universe 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 is a set of elements from some universe , then the complement of is the set

$\displaystyle 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 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 is the universe of
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:

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

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.

"union" is owned by rm50. [ full author list (3) | owner history (2) ]

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