# Intersection (set theory)

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]

Type The intersection of two sets ${\displaystyle A}$ and ${\displaystyle B,}$ represented by circles. ${\displaystyle A\cap B}$ is in red. Set operation Set theory 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 ${\displaystyle B}$. ${\displaystyle A\cap B=\{x:x\in A{\text{ and }}x\in B\}}$

## Notation and terminology

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

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

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

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

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

The intersection of more than two sets (generalized intersection) can be written as:
${\displaystyle \bigcap _{i=1}^{n}A_{i}}$

which is similar to capital-sigma notation.

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

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

Thus the parentheses may be omitted without ambiguity: either of the above can be written as ${\displaystyle A\cap B\cap C}$ . Intersection is also commutative. That is, for any ${\displaystyle A}$  and ${\displaystyle B,}$  one has
${\displaystyle A\cap B=B\cap A.}$

The intersection of any set with the empty set results in the empty set; that is, that for any set ${\displaystyle A}$ ,
${\displaystyle A\cap \varnothing =\varnothing }$

Also, the intersection operation is idempotent; that is, any set ${\displaystyle A}$  satisfies that ${\displaystyle A\cap A=A}$ . All these properties follow from analogous facts about logical conjunction.

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}}}

Inside a universe ${\displaystyle U,}$  one may define the complement ${\displaystyle A^{c}}$  of ${\displaystyle A}$  to be the set of all elements of ${\displaystyle U}$  not in ${\displaystyle A.}$  Furthermore, the intersection of ${\displaystyle A}$  and ${\displaystyle B}$  may be written as the complement of the union of their complements, derived easily from De Morgan's laws:
${\displaystyle A\cap B=\left(A^{c}\cup B^{c}\right)^{c}}$

## 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)

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

If ${\displaystyle M}$  is empty, there are no sets ${\displaystyle A}$  in ${\displaystyle M,}$  so the question becomes "which ${\displaystyle x}$ 's satisfy the stated condition?" The answer seems to be every possible ${\displaystyle x}$ . When ${\displaystyle M}$  is empty, the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set (the identity element for the operation of intersection),[4] but in standard (ZF) set theory, the universal set does not exist.

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 ${\displaystyle \tau }$ ).