withineither

Withineither is a relational concept used to express that a value lies within at least one of two specified sets or intervals. The term combines within (containment) with either (one or the other), to denote disjunctive containment. It is used in mathematics, computer science, and constraint systems to simplify conditional checks.

Formally, let A and B be subsets of a universal set U. A value x satisfies withineither(A,B) if x ∈ A or x ∈ B (or both). If A and B are closed intervals on the real line, A=[a,b] and B=[c,d], then x satisfies withineither if a ≤ x ≤ b or c ≤ x ≤ d. In implementation, the endpoints are often treated inclusively, but the convention may vary.

In programming, withineither is implemented as a logical disjunction: withineither(x, A, B) = (x ∈ A) || (x ∈ B). When A and B are ranges, this is equivalent to the union of the ranges, in the sense that x is in the union.

Example: x=0.5 is withineither [0,1] and [2,3]? It's in the first. x=2.5 is in the second. x=1.5 is not in either.

Applications include data validation, rule engines, and search queries. If A and B overlap, withineither reduces to testing membership in the union A ∪ B; care is needed when further distinctions are required between exclusivity and inclusivity.

See also: union, disjunction, inclusive or, range checking.