nonintersecting

Nonintersecting is an adjective used to describe two or more objects that have no points in common; in other words, their intersection is empty. The term is used in geometry, set theory, and combinatorics to express that objects do not meet or overlap.

Geometric sense: For lines in the Euclidean plane, nonintersecting lines are parallel, since two nonparallel lines meet at a point. In three-dimensional space, two lines can be nonintersecting yet not parallel, a situation known as skew lines. Curves or surfaces can be nonintersecting if they do not meet anywhere.

Set-theoretic and probabilistic sense: A family of sets is nonintersecting or disjoint if every pair has empty intersection. If A and B are nonintersecting, A ∩ B = ∅. In probability, disjoint events are mutually exclusive, meaning they cannot occur simultaneously.

In graph theory and combinatorics: a collection of nonintersecting paths or curves often means the paths are vertex-disjoint, sharing no vertices; sometimes edge-disjointness is also specified. The concept is central to results involving families of nonintersecting paths, such as the Lindström–Gessel–Viennot framework.

Related terms include noncrossing, which concerns the nonappearance of intersections in a particular drawing or ordering, and mutual exclusivity in statistics. In all contexts, nonintersecting emphasizes that objects occupy separate points or regions of the ambient space.