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 } , {\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.

## 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 ) {\displaystyle (A\cap B)\cap C=A\cap (B\cap C)} ;
* commutative :  A ∩ B = B ∩ A {\displaystyle A\cap B=B\cap A} .

## General intersections[edit]

### Finite intersections[edit]

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

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

### 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 ∈ Λ X λ = { x : ∀ λ ∈ Λ , x ∈ X λ } . {\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:

⋂ X = { x : ∀ Y ∈ X , x ∈ Y } . {\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

A ∩ B = ⋂ { A , B } . {\displaystyle A\cap B=\bigcap \\{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.