The intersection of $A$ and $B$ is the set $A\cap B$ of elements that lie in both set $A$ and set $B$.
Symbolic statement
$A\cap B=\{x:x\in A{\text{ and }}x\in B\}$
In
set theory, the intersection of two
sets$A$ and $B,$ denoted by $A\cap B,$^{
[1]} is the set containing all elements of $A$ that also belong to $B$ or equivalently, all elements of $B$ that also belong to $A.$^{
[2]}
Notation and terminology
Intersection is written using the symbol "$\cap$" between the terms; that is, in
infix notation. For example:
The intersection of two sets $A$ and $B,$ denoted by $A\cap B$,^{
[3]} is the set of all objects that are members of both the sets $A$ and $B.$
In symbols:
$A\cap B=\{x:x\in A{\text{ and }}x\in B\}.$
That is, $x$ is an element of the intersection $A\cap B$if and only if$x$ is both an element of $A$ and an element of $B.$^{
[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 $A$ intersects (meets) $B$ if there exists some $x$ that is an element of both $A$ and $B,$ in which case we also say that $A$ intersects (meets) $B$at$x$. Equivalently, $A$ intersects $B$ if their intersection $A\cap B$ is an inhabited set, meaning that there exists some $x$ such that $x\in A\cap B.$
We say that $A$ and $B$ are
disjoint if $A$ does not intersect $B.$ In plain language, they have no elements in common. $A$ and $B$ are disjoint if their intersection is
empty, denoted $A\cap B=\varnothing .$
For example, the sets $\{1,2\}$ and $\{3,4\}$ are disjoint, while the set of even numbers intersects the set of
multiples of 3 at the multiples of 6.
Binary intersection is an
associative operation; that is, for any sets $A,B,$ and $C,$ one has
$A\cap (B\cap C)=(A\cap B)\cap C.$
Thus the parentheses may be omitted without ambiguity: either of the above can be written as $A\cap B\cap C$. Intersection is also
commutative. That is, for any $A$ and $B,$ one has
$A\cap B=B\cap A.$
The intersection of any set with the
empty set results in the empty set; that is, that for any set $A$,
$A\cap \varnothing =\varnothing$
Also, the intersection operation is
idempotent; that is, any set $A$ satisfies that $A\cap A=A$. All these properties follow from analogous facts about
logical conjunction.
Intersection
distributes over
union and union distributes over intersection. That is, for any sets $A,B,$ and $C,$ one has
Inside a universe $U,$ one may define the
complement$A^{c}$ of $A$ to be the set of all elements of $U$ not in $A.$ Furthermore, the intersection of $A$ and $B$ may be written as the complement of the
union of their complements, derived easily from
De Morgan's laws:
The most general notion is the intersection of an arbitrary nonempty collection of sets.
If $M$ is a
nonempty set whose elements are themselves sets, then $x$ is an element of the intersection of $M$ if and only if
for every element $A$ of $M,$$x$ is an element of $A.$
In symbols:
The notation for this last concept can vary considerably.
Set theorists will sometimes write "$\bigcap M$", while others will instead write "${\bigcap }_{A\in M}A$".
The latter notation can be generalized to "${\bigcap }_{i\in I}A_{i}$", which refers to the intersection of the collection $\left\{A_{i}:i\in I\right\}.$
Here $I$ is a nonempty set, and $A_{i}$ is a set for every $i\in I.$
When formatting is difficult, this can also be written "$A_{1}\cap A_{2}\cap A_{3}\cap \cdots$". 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 $M$ was the
empty set ($\varnothing$). The reason is as follows: The intersection of the collection $M$ is defined as the set (see
set-builder notation)
$\bigcap _{A\in M}A=\{x:{\text{ for all }}A\in M,x\in A\}.$
If $M$ is empty, there are no sets $A$ in $M,$ so the question becomes "which $x$'s satisfy the stated condition?" The answer seems to be every possible $x$. When $M$ 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 $X$, the notion of the intersection of an empty collection of subsets of $X$ is well-defined. In that case, if $M$ is empty, its intersection is $\bigcap M=\bigcap \varnothing =\{x\in X:x\in A{\text{ for all }}A\in \varnothing \}$. Since all $x\in X$ vacuously satisfy the required condition, the intersection of the empty collection of subsets of $X$ is all of $X.$ In formulas, $\bigcap \varnothing =X.$ 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$x$ is of a prescribed type $\tau ,$ so the intersection is understood to be of type $\mathrm {set} \ \tau$ (the type of sets whose elements are in $\tau$), and we can define $\bigcap _{A\in \emptyset }A$ to be the universal set of $\mathrm {set} \ \tau$ (the set whose elements are exactly all terms of type $\tau$).
See also
Algebra of sets – Identities and relationships involving sets
Cardinality – Definition of the number of elements in a set
Complement – Set of the elements not in a given subset
Intersection (Euclidean geometry) – Shape formed from points common to other shapesPages displaying short descriptions of redirect targets
Intersection graph – Graph representing intersections between given sets