Intersection

The term intersection refers to the point or set of points common to two or more objects. In geometry and everyday language it denotes where lines or curves cross; in set theory it denotes the common elements of sets.

In Euclidean geometry, the intersection of two figures is the set of points contained in both. The intersection of two lines in a plane is a single point if the lines are not parallel; there is no intersection if they are parallel and distinct; and if they coincide, the intersection is the entire line. The intersection operator is commutative and associative for sets: A ∩ B = B ∩ A, (A ∩ B) ∩ C = A ∩ (B ∩ C).

In set theory and probability, A ∩ B denotes the elements that belong to both A and B. For probabilities, P(A ∩ B) is the probability that both events occur; relationships such as independence influence its value. In Venn diagrams the intersection is shown as the overlapping region.

In topology, the intersection of open sets is open for any finite collection; infinite intersections of open sets need not be open. The concept generalizes to intersections over arbitrary index sets when defining topological structures.

In daily language, intersection also refers to a road junction where two or more roads cross or meet, often controlled by traffic signals or signs. Types include at-grade intersections and grade-separated interchanges.