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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 $A_{j}$:
$\bigcap_{{j\in J}}A_{j}=\{x:x\in A_{j}\mbox{ for all }j\in J\}.$ 
A set $U$ intersects, or meets, a set $V$ if $U\cap V$ is nonempty.
Some elementary properties of $\cap$ are

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

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

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

$A\cap A^{\complement}=\varnothing$, where $A^{\complement}$ is the complement of $A$ in some fixed universe $U$.
Remark. What is $\bigcap_{{j\in J}}A_{j}$ when $J=\varnothing$? 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 $U$.
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