Definition – Set intersection.

Definition

Let $A, B$ be sets. The intersection of two sets, written $A\cap{B}$ , is defined as,

$A\cap{B}$ = $\{x~|~x\in{A}~\wedge~x\in{B}\}$

We can extend this idea to the intersection of three or more sets. Given a family of sets $\mathcal{F}$ , we write the intersection of all sets in $\mathcal{F}$ as

$\bigcap\limits_{i} \mathcal{F}_{i}$

As long as $\mathcal{F}$ has a finite number of sets. If, however, the family of sets is either infinite, or we are unable to assign an index to each of the sets within $\mathcal{F}$ , then we can still write the intersection of all the sets in $\mathcal{F}$ as follows,

$\bigcap\limits_{B\in \mathcal{F}} B_{i} = \{x~|\not\exists{B\in{\mathcal{F}}}~s.t~x\notin{B}\}$

Notes

Ie, if $x$ is not in every set contained in $\mathcal{F}$ , then we do not include it in the intersection.

Set intersections obey the laws of distributivity, commutativity and distributivity. A proof of these properties is given here.