Intersects

Intersects is the third-person singular form of the verb intersect, meaning to cross or to share common elements. In mathematics and related fields, the term describes either the act of crossing in a geometric sense or the relation of sharing members between sets.

In set theory, the intersection of two sets A and B, written A ∩ B, is the set of all elements that belong to both A and B. Intersection is commutative (A ∩ B = B ∩ A) and associative (A ∩ (B ∩ C) = (A ∩ B) ∩ C). It is idempotent (A ∩ A = A) and, with the universal set U, satisfies A ∩ U = A; the intersection can be empty, yielding the empty set Ø.

In geometry, the intersection of objects is the set of points common to them. Two lines in a plane intersect at a point unless they are parallel; a line and a plane may intersect at a point or the line may lie entirely in the plane; two planes intersect in a line. Intersection describes where geometric figures meet and is used to study position, overlap, and containment.

In probability and statistics, the intersection of events A and B is A ∩ B, the outcome that both occur. Probabilities satisfy P(A ∩ B) ≤ min{P(A), P(B)}, with equality if A and B are independent.

In topology, the intersection of a collection of open sets need not be open in general, though the intersection of closed sets is closed. In computer science and geographic information systems, intersects is used as a predicate to test whether two shapes share any points, underpinning spatial queries and indexes.