# Intersection

(Redirected from Intersecting)
Jump to navigationJump to search ## Origin

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

## Definitions

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.

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