Narrowerthan

Narrowerthan is a binary relation used in mathematics, computer science, and related fields to compare the width of two objects. It is defined relative to a specified width function that assigns a non-negative width value to each object in a domain. Under this definition, x is narrowerthan than y if and only if the width of x is strictly less than the width of y (w(x) < w(y)).

Formal definition and properties

Let D be a set of objects and w: D -> R+ be a width function that is well-defined for all elements of D. The relation narrowerthan on D is the set of pairs (x, y) such that w(x) < w(y). This makes narrowerthan a strict order: it is irreflexive (no element is narrower than itself) and transitive (if w(x) < w(y) and w(y) < w(z), then w(x) < w(z)). If widths can be equal, some pairs will be incomparable under this strict relation, so narrowerthan is not necessarily a total order in general.

Domain and usage

Narrowerthan is used in scenarios where a single dimensional extent serves as a criterion for comparison. Common domains include geometric shapes (comparing widths of rectangles or bounding boxes), user interface layouts (element widths in a responsive grid), typography (character or glyph width), and data visualization (bar or column widths). In programming, it may be implemented as a predicate or function such as narrowerthan(x, y) that evaluates to true when w(x) < w(y).

Variations and related concepts

Width is the chosen metric, and different definitions of width (bounding box width, diameter, major axis length, etc.) yield different instantiations of narrowerthan. Related relations include wider-than and narrower-or-equal, the latter combining the strict comparison with equality.

See also

wider-than, width (geometry), order relations, comparison operators.