# Intersection

From Citizendium

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

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

## Contents

## Properties

The intersection operation is:

- associative : ;
- commutative : .

## General intersections

### Finite intersections

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

### Infinite intersections

The intersection of a general family of sets *X*_{λ} as λ ranges over a general index set Λ may be written in similar notation as

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:

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

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

## See also

## References

- 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.