Intersection

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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∩B={x:x∈A∧x∈B},

where ∧ denotes logical and. Two sets are disjoint if their intersection is the empty set.

## Contents

* 1 Properties
* 2 General intersections
* 2.1 Finite intersections
* 2.2 Infinite intersections
* 3 See also
* 4 References

## Properties[edit]

The intersection operation is:

* associative : (A∩B)∩C=A∩(B∩C);
* commutative : A∩B=B∩A.

## General intersections[edit]

### Finite intersections[edit]

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

⋂i=1nXi=X1∩(X2∩(X3∩(⋯Xn)⋯))).

### Infinite intersections[edit]

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

⋂λ∈ΛXλ={x:∀λ∈Λ,x∈Xλ}.

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:

⋂X={x:∀Y∈X,x∈Y}.

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

A∩B=⋂{A,B}.

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

## See also[edit]

* Union

## References[edit]

* Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold. Section 4.
* Keith J. Devlin (1979). Fundamentals of Contemporary Set Theory. Springer-Verlag, 6,11. ISBN 0-387-90441-7.