Intervaltype

Intervaltype is a data concept used to represent a contiguous range of elements within a totally ordered set, most often the real numbers. It encodes an interval by storing its bounds and, optionally, whether each bound is included or excluded. An interval may be bounded (both ends finite) or unbounded (one or both ends extend to infinity).

A typical interval is described by a lower bound L and an upper bound U, with flags indicating whether L and/or U are closed (included) or open (excluded). Common forms are the closed interval [L, U], the open interval (L, U), and the half-open intervals [L, U) and (L, U]. A degenerate interval occurs when L = U, representing a single point, while an empty interval denotes no elements.

Operations commonly defined for intervaltype include membership testing (whether a value x lies in the interval), intersection and union with other intervals, and checks for disjoint or overlapping intervals. In numerical contexts, interval arithmetic extends these notions to compute bounds of expressions, for example adding [a, b] and [c, d] yields [a + c, b + d], with similar rules for subtraction, multiplication, and division, taking care with sign and endpoint openness.

Applications span mathematics, computer science, and engineering. Intervaltype supports constraint programming, scheduling and temporal reasoning, uncertainty propagation, and robust numerical analysis where exact values are unknown but bounded within intervals. Implementations often normalize intervals to ensure L ≤ U (or represent empty intervals explicitly), and provide efficient tests for containment, intersection, and emptiness. Endpoints can be finite or infinite, allowing representations like [a, ∞), (-∞, b], or (-∞, ∞) as appropriate.