intersect

Intersect is a verb meaning to cross or meet at a point or along a line. In mathematics, the term appears in several contexts. In geometry, two lines intersect if they share a common point; they may intersect at a single point, be parallel (no intersection), or coincide (infinitely many intersection points). In set theory, the intersection of sets A and B, written A ∩ B, consists of all elements that belong to both sets; the intersection is empty when the sets have no elements in common. The intersection operation is commutative and associative.

More generally, the concept extends to time intervals, events, and other objects. Two time intervals intersect if there exists a moment that lies in both; in probability, the intersection of events refers to outcomes that satisfy all involved events. In topology and analysis, the intersection of open sets remains open, and the intersection of subspaces is defined analogously to the set-theoretic case.

In geometry, terms such as intersecting lines, the intersection point, and the line of intersection (for planes in three-dimensional space) are commonly used. In computational contexts, intersection operations appear in algorithms for graphics, collision detection, and spatial queries, as well as in database systems, where results sets can be intersected to yield common elements.

Etymology: intersect derives from Latin intersecāre, from inter- “between” and secāre “to cut,” reflecting the idea of cutting across or meeting at points of contact.