# Intersection

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

${\displaystyle A\cap B=\{x:x\in A\wedge x\in B\},\,}$

where ${\displaystyle \wedge }$ denotes logical and. Two sets are disjoint if their intersection is the empty set.

## Properties

The intersection operation is:

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

## General intersections

### Finite intersections

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

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

### 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

${\displaystyle \bigcap _{\lambda \in \Lambda }X_{\lambda }=\{x:\forall \lambda \in \Lambda ,~x\in X_{\lambda }\}.\,}$

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:

${\displaystyle \bigcap X=\{x:\forall Y\in X,~x\in Y\}.\,}$

In this notation the intersection of two sets A and B may be expressed as

${\displaystyle A\cap B=\bigcap \{A,B\}.\,}$

The correct definition of the intersection of the empty set needs careful consideration.