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 nonempty.
Some elementary properties of $\cap $ are

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(idempotency) $A\cap A=A$,

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(commutativity) $A\cap B=B\cap A$,

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

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$A\cap {A}^{\mathrm{\complement}}=\mathrm{\varnothing}$, 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 NeumannGödelBernays) 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  20130322 12:14:52 
Last modified on  20130322 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 