# Intersection

## Origin

Latin *intersectiōn-em* (Vitruvius), n. of action from *intersecāre*

## Definitions

- 1: The action or fact of intersecting or crossing
- 2:a. The place where two things intersect or cross, spec.cross-road

- b. Geometry, the point (or line) of intersection; the point common to two lines or a line and a surface (or the line common to two surfaces) which intersect.

- 3:a. Logic. The relation of two classes that intersect, i.e. each of which partly includes and partly excludes the other.

- b. Logic and Mathematics. The set which comprises all the elements common to two or more given sets, and no others; also, the operation of forming such a set.

## Description

In mathematics, the intersection (denoted as ∩) of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

The intersection of A and B is written "A ∩ B". Formally:

- x ∈ A ∩ B if and only if
- x ∈ A and
- x ∈ B.

- 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, …}.

If the intersection of two sets A and B is empty, that is they have no elements in common, then they are said to be disjoint, denoted: A ∩ B = ∅. For example the sets {1, 2} and {3, 4} are disjoint, written {1, 2} ∩ {3, 4} = ∅.

More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an associative operation; thus, A ∩ (B ∩ C) = (A ∩ B) ∩ C.

If the sets A and B are closed under complement then 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: A ∩ B = (Ac ∪ Bc)c[1]