Intersection

The intersection of two sets ${\displaystyle A}$ and ${\displaystyle B,}$ represented by circles. ${\displaystyle A\cap B}$ is in red.
Type	Set operation
Field	Set theory
Statement	The intersection of ${\displaystyle A}$ and ${\displaystyle B}$ is the set ${\displaystyle A\cap B}$ of elements that lie in both set ${\displaystyle A}$ and set ${\displaystyle B}$.
Symbolic statement	${\displaystyle A\cap B=\{x:x\in A{\text{ and ))x\in B\))$

In set theory, the intersection of two sets ${\displaystyle A}$ and ${\displaystyle B,}$ denoted by ${\displaystyle A\cap B,}$^[1] is the set containing all elements of ${\displaystyle A}$ that also belong to ${\displaystyle B}$ or equivalently, all elements of ${\displaystyle B}$ that also belong to ${\displaystyle A.}$^[2]

Notation and terminology

Intersection is written using the symbol "${\displaystyle \cap }$" between the terms; that is, in infix notation. For example:

$${\displaystyle \{1,2,3\}\cap \{2,3,4\}=\{2,3\))$$

$${\displaystyle \{1,2,3\}\cap \{4,5,6\}=\varnothing }$$

$${\displaystyle \mathbb {Z} \cap \mathbb {N} =\mathbb {N} }$$

$${\displaystyle \{x\in \mathbb {R} :x^{2}=1\}\cap \mathbb {N} =\{1\))$$

$${\displaystyle \bigcap _{i=1}^{n}A_{i))$$

For an explanation of the symbols used in this article, refer to the table of mathematical symbols.

Definition

The intersection of two sets ${\displaystyle A}$ and ${\displaystyle B,}$ denoted by ${\displaystyle A\cap B}$,^[3] is the set of all objects that are members of both the sets ${\displaystyle A}$ and ${\displaystyle B.}$ In symbols:

$${\displaystyle A\cap B=\{x:x\in A{\text{ and ))x\in B\}.}$$

That is, ${\displaystyle x}$ is an element of the intersection ${\displaystyle A\cap B}$ if and only if ${\displaystyle x}$ is both an element of ${\displaystyle A}$ and an element of ${\displaystyle B.}$^[3]

For example:

• The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.
• The number 9 is not in the intersection of the set of prime numbers {2, 3, 5, 7, 11, ...} and the set of odd numbers {1, 3, 5, 7, 9, 11, ...}, because 9 is not prime.

Intersecting and disjoint sets

We say that ${\displaystyle A}$ intersects (meets) ${\displaystyle B}$ if there exists some ${\displaystyle x}$ that is an element of both ${\displaystyle A}$ and ${\displaystyle B,}$ in which case we also say that ${\displaystyle A}$ intersects (meets) ${\displaystyle B}$ at ${\displaystyle x}$. Equivalently, ${\displaystyle A}$ intersects ${\displaystyle B}$ if their intersection ${\displaystyle A\cap B}$ is an inhabited set, meaning that there exists some ${\displaystyle x}$ such that ${\displaystyle x\in A\cap B.}$

We say that ${\displaystyle A}$ and ${\displaystyle B}$ are disjoint if ${\displaystyle A}$ does not intersect ${\displaystyle B.}$ In plain language, they have no elements in common. ${\displaystyle A}$ and ${\displaystyle B}$ are disjoint if their intersection is empty, denoted ${\displaystyle A\cap B=\varnothing .}$

For example, the sets ${\displaystyle \{1,2\))$ and ${\displaystyle \{3,4\))$ are disjoint, while the set of even numbers intersects the set of multiples of 3 at the multiples of 6.

Algebraic properties

Binary intersection is an associative operation; that is, for any sets ${\displaystyle A,B,}$ and ${\displaystyle C,}$ one has

$${\displaystyle A\cap (B\cap C)=(A\cap B)\cap C.}$$

${\displaystyle A\cap B\cap C}$

${\displaystyle A}$

${\displaystyle B,}$

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

${\displaystyle A}$

$${\displaystyle A\cap \varnothing =\varnothing }$$

${\displaystyle A}$

${\displaystyle A\cap A=A}$

Intersection distributes over union and union distributes over intersection. That is, for any sets ${\displaystyle A,B,}$ and ${\displaystyle C,}$ one has

$${\displaystyle {\begin{aligned}A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\\A\cup (B\cap C)=(A\cup B)\cap (A\cup C)\end{aligned))}$$

${\displaystyle U,}$

${\displaystyle A^{c))$

${\displaystyle A}$

${\displaystyle U}$

${\displaystyle A.}$

${\displaystyle A}$

${\displaystyle B}$

$${\displaystyle A\cap B=\left(A^{c}\cup B^{c}\right)^{c))$$

Arbitrary intersections

The most general notion is the intersection of an arbitrary nonempty collection of sets. If ${\displaystyle M}$ is a nonempty set whose elements are themselves sets, then ${\displaystyle x}$ is an element of the intersection of ${\displaystyle M}$ if and only if for every element ${\displaystyle A}$ of ${\displaystyle M,}$ ${\displaystyle x}$ is an element of ${\displaystyle A.}$ In symbols:

$${\displaystyle \left(x\in \bigcap _{A\in M}A\right)\Leftrightarrow \left(\forall A\in M,\ x\in A\right).}$$

The notation for this last concept can vary considerably. Set theorists will sometimes write "${\displaystyle \bigcap M}$", while others will instead write "${\displaystyle {\bigcap }_{A\in M}A}$". The latter notation can be generalized to "${\displaystyle {\bigcap }_{i\in I}A_{i))$", which refers to the intersection of the collection ${\displaystyle \left\{A_{i}:i\in I\right\}.}$ Here ${\displaystyle I}$ is a nonempty set, and ${\displaystyle A_{i))$ is a set for every ${\displaystyle i\in I.}$

In the case that the index set ${\displaystyle I}$ is the set of natural numbers, notation analogous to that of an infinite product may be seen:

$${\displaystyle \bigcap _{i=1}^{\infty }A_{i}.}$$

When formatting is difficult, this can also be written "${\displaystyle A_{1}\cap A_{2}\cap A_{3}\cap \cdots }$". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras.

Nullary intersection

In the previous section, we excluded the case where ${\displaystyle M}$ was the empty set (${\displaystyle \varnothing }$). The reason is as follows: The intersection of the collection ${\displaystyle M}$ is defined as the set (see set-builder notation)

$${\displaystyle \bigcap _{A\in M}A=\{x:{\text{ for all ))A\in M,x\in A\}.}$$

${\displaystyle M}$

${\displaystyle A}$

${\displaystyle M,}$

${\displaystyle x}$

every possible ${\displaystyle x}$

${\displaystyle M}$

However, when restricted to the context of subsets of a given fixed set ${\displaystyle X}$, the notion of the intersection of an empty collection of subsets of ${\displaystyle X}$ is well-defined. In that case, if ${\displaystyle M}$ is empty, its intersection is ${\displaystyle \bigcap M=\bigcap \varnothing =\{x\in X:x\in A{\text{ for all ))A\in \varnothing \))$. Since all ${\displaystyle x\in X}$ vacuously satisfy the required condition, the intersection of the empty collection of subsets of ${\displaystyle X}$ is all of ${\displaystyle X.}$ In formulas, ${\displaystyle \bigcap \varnothing =X.}$ This matches the intuition that as collections of subsets become smaller, their respective intersections become larger; in the extreme case, the empty collection has an intersection equal to the whole underlying set.

Also, in type theory ${\displaystyle x}$ is of a prescribed type ${\displaystyle \tau ,}$ so the intersection is understood to be of type ${\displaystyle \mathrm {set} \ \tau }$ (the type of sets whose elements are in ${\displaystyle \tau }$), and we can define ${\displaystyle \bigcap _{A\in \emptyset }A}$ to be the universal set of ${\displaystyle \mathrm {set} \ \tau }$ (the set whose elements are exactly all terms of type ${\displaystyle \tau }$).