intersection

The intersection of two sets $A$ and $B$ is the set that contains all the elements $x$ such that $x\in A$ and $x\in B$. The intersection of $A$ and $B$ is written as $A\cap B$. The following Venn diagram 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 $A\cap B=\{1,3,5\}$.

We can also define the intersection of an arbitrary number of sets. If ${\{A_j\}}_{j\in J}$ is a family of sets, we define the intersection of all them, denoted ${\bigcap }_{j\in J}{A}_j$, as the set consisting of those elements belonging to every set $A_j$:

$$\bigcap _{j\in J}{A}_j=\{x:x\in A_j\text{ for all }j\in J\}.$$

A set $U$ intersects, or meets, a set $V$ if $U\cap V$ is non-empty.

Some elementary properties of $\cap$ are

• (idempotency) $A\cap A=A$,

• (commutativity) $A\cap B=B\cap A$,

• (associativity) $A\cap (B\cap C)=(A\cap B)\cap C$,

• $A\cap A^{\mathrm{\complement }}=\mathrm{\varnothing }$, where $A^{\mathrm{\complement }}$ is the complement of $A$ in some fixed universe $U$.

Remark. What is ${\bigcap }_{j\in J}{A}_j$ when $J=\mathrm{\varnothing }$? In other words, what is the intersection of an empty family of sets? First note that if $I\subseteq J$, then

$$\bigcap _{j\in J}{A}_j\subseteq \bigcap _{i\in I}{A}_i.$$

This leads the conclusion that the intersection of an empty family of sets should be as large as possible. How large should it be? In addition, is this intersection a set? The answer depends on what versions of set theory 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 $A_j\subseteq U$, then it is sometimes a matter of convenience to define the intersection of an empty family of $A_j$ 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