The intersectionMathworldPlanetmathPlanetmath of two sets A and B is the set that contains all the elements x such that xA and xB. The intersection of A and B is written as AB. The following Venn diagramMathworldPlanetmath illustrates the intersection of two sets A and B:


Example. If A={1,2,3,4,5} and B={1,3,5,7,9} then AB={1,3,5}.

We can also define the intersection of an arbitrary number of sets. If {Aj}jJ is a family of sets, we define the intersection of all them, denoted jJAj, as the set consisting of those elements belonging to every set Aj:

jJAj={x:xAj for all jJ}.

A set U intersects, or meets, a set V if UV is non-empty.

Some elementary properties of are

Remark. What is jJAj when J=? In other words, what is the intersection of an empty family of sets? First note that if IJ, then


This leads the conclusionMathworldPlanetmath that the intersection of an empty family of sets should be as large as possible. How large should it be? In additionPlanetmathPlanetmath, is this intersection a set? The answer depends on what versions of set theoryMathworldPlanetmath we are working in. Some theories (for example, von Neumann-Gödel-Bernays) say this is the class V of all sets, while others do not define this notion at all. However, if there is a fixed set U in advance such that each AjU, then it is sometimes a matter of convenience to define the intersection of an empty family of Aj to be U.

Title intersection
Canonical name Intersection
Date of creation 2013-03-22 12:14:52
Last modified on 2013-03-22 12:14:52
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 22
Author CWoo (3771)
Entry type Definition
Classification msc 03E99
Synonym intersects
Synonym meets
Related topic union
Related topic Union
Related topic FiniteIntersectionProperty
Related topic EmptySet
Related topic ProductOfLeftAndRightIdeal