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 $ℝ$, 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

$$A^C=U\backslash \backslash A=\{x\in U\mid x\notin A\}$$

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:

$$\mathrm{\{}1,2\}\cup \{3,4\}=\{1,2,3,4\}$$

$$\mathrm{\{}blue,green\}\cup \mathrm{\varnothing }=\{blue,green\}$$

$$\mathrm{\{}x\in \mathbb{Z}\mid x⩾1\}\cup \{x\in \mathbb{Z}\mid x⩽-1\}=\{x\in \mathbb{Z}\mid x\ne 0\}$$

$$\mathrm{\{}1,2\}\cup \{1,4\}=\{1,2,4\}$$

$$\mathrm{\{}x\in ℝ\mid x⩾2\}\cup \{x\in ℝ\mid x⩽2\}=ℝ$$

$$\bigcup _{\begin{array}{c}n\in \mathbb{Z}\\ n>0\end{array}}\{\frac{p}{n}\mid p\in \mathbb{Z}\}=\mathbb{Q}$$

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.