# Intersection (set theory)

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In mathematics, the intersection of two sets A and B, denoted by AB, [1] [2] is the set containing all elements of A that also belong to B (or equivalently, all elements of B that also belong to A). [3]

## Notation and terminology

Intersection is written using the sign "∩" between the terms; that is, in infix notation. For example,

${\displaystyle \{1,2,3\}\cap \{2,3,4\}=\{2,3\}}$
${\displaystyle \{1,2,3\}\cap \{4,5,6\}=\emptyset }$
${\displaystyle \mathbb {Z} \cap \mathbb {N} =\mathbb {N} }$
${\displaystyle \{x\in \mathbb {R} :x^{2}=1\}\cap \mathbb {N} =\{1\}}$

The intersection of more than two sets (generalized intersection) can be written as [1]

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

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 A and B, denoted by AB, [1] [4] is the set of all objects that are members of both the sets A and B. In symbols,

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

That is, x is an element of the intersection AB, if and only if x is both an element of A and an element of B. [4]

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.

Intersection is an associative operation; that is, for any sets A, B, and C, one has A ∩ (BC) = (AB) ∩ C. Intersection is also commutative; for any A and B, one has AB = BA. It thus makes sense to talk about intersections of multiple sets. The intersection of A, B, C, and D, for example, is unambiguously written ABCD.

Inside a universe U, one may define the complement Ac 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:
AB = (AcBc)c

### Intersecting and disjoint sets

We say that A intersects (meets) B at an element x if x belongs to A and B. We say that A intersects (meets) B if A intersects B at some element. A intersects B if their intersection is inhabited.

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

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.

## Arbitrary intersections

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:

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

The notation for this last concept can vary considerably. Set theorists will sometimes write "⋂M", while others will instead write "⋂AM A". The latter notation can be generalized to "⋂iI Ai", which refers to the intersection of the collection {Ai : i  I}. Here I is a nonempty set, and Ai is a set for every i in I.

In the case that the index set 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 "A1 A2 A3 ∩ ...". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras.

## Nullary intersection

Note that in the previous section, we excluded the case where M was the empty set (∅). The reason is as follows: The intersection of the collection M is defined as the set (see set-builder notation)

${\displaystyle \bigcap _{A\in M}A=\{x:\forall A\in M,x\in A\}.}$

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), [5] but in standard (ZFC) set theory, the universal set does not exist.

In type theory however, 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 ${\displaystyle \tau }$).

## Related Research Articles

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