# Intersection  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

In set theory, the intersection of two sets is the set of elements that they have in common:

$A\cap B=\{x:x\in A\wedge x\in B\},\,$ where $\wedge$ denotes logical and. Two sets are disjoint if their intersection is the empty set.

## Properties

The intersection operation is:

• associative : $(A\cap B)\cap C=A\cap (B\cap C)$ ;
• commutative : $A\cap B=B\cap A$ .

## General intersections

### Finite intersections

The intersection of any finite number of sets may be defined inductively, as

$\bigcap _{i=1}^{n}X_{i}=X_{1}\cap (X_{2}\cap (X_{3}\cap (\cdots X_{n})\cdots ))).\,$ ### Infinite intersections

The intersection of a general family of sets Xλ as λ ranges over a general index set Λ may be written in similar notation as

$\bigcap _{\lambda \in \Lambda }X_{\lambda }=\{x:\forall \lambda \in \Lambda ,~x\in X_{\lambda }\}.\,$ We may drop the indexing notation and define the intersection of a set to be the set of elements contained in all the elements of that set:

$\bigcap X=\{x:\forall Y\in X,~x\in Y\}.\,$ In this notation the intersection of two sets A and B may be expressed as

$A\cap B=\bigcap \{A,B\}.\,$ The correct definition of the intersection of the empty set needs careful consideration.