The union of two sets A and B is the set which contains all xA and all xB, denoted AB. In the Venn diagramMathworldPlanetmath below, AB is the entire area shaded in blue.


We can extend this to any (finite or infiniteMathworldPlanetmath) family (Ai)iI, writing iIAi for the union of this family. Formally, for a family (Ai)iI of sets:


Alternatively, and equivalently,

xiIAiiI such that xAi

This characterizationMathworldPlanetmath makes it much clearer that if I is itself the empty setMathworldPlanetmath (that is, if we are taking the union of an empty family), then the union is empty; that is,


Often elements of sets are taken from some universePlanetmathPlanetmath U of elements under consideration (for example, the real numbers , 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


From an axiomatic point of view, the existence of the union is guaranteed by the axiom of union.

Note that the sets Ai may be, but need not be, disjoint. Unions satisfy some basic properties that are obvious from the definitions:

Here are some examples of set unions:


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 IntersectionDlmfMathworldPlanetmath