variablesuch

Variablesuch is a term used to describe a variable that is constrained by a predicate introduced with the phrase such that. In many mathematical writings, a variable is said to be such that when its admissible values are restricted by a condition, for example x such that x > 0. The concept can be formalized as a pair consisting of a variable and a predicate. In set-builder notation, { x in D | P(x) } describes all values of x that satisfy P; a variablesuch viewpoint highlights x as the value chosen under the constraint P.

In computer science, especially in constraint programming and logic programming, variables are associated with domains and constraints; a variable described as such that P(v) holds when the chosen value satisfies the predicate. The idea is common when describing solution spaces or search problems and when expressing conditions that define admissible assignments.

Examples help illustrate the idea: Let x be a variable such that x^2 < 9 with integer domain; the values that satisfy the constraint are -2, -1, 0, 1, 2. The variablesuch perspective emphasizes the role of the condition in selecting permissible values, rather than merely naming a variable.

Relation to existential quantification: Such-that clauses express a condition on the variable rather than asserting the mere existence of some x with P(x). The phrase x such that P(x) names a value that must satisfy P, whereas an existential statement (there exists x such that P(x)) asserts existence without naming the value.

Terminology and usage: The term variablesuch is not standard in formal mathematics; most texts prefer x such that or set-builder notation. It is mainly used as a descriptive or pedagogical label in discussions of constrained variables.

See also: set-builder notation, existential quantification, constraint programming, logic programming, predicate.