relationsfor

Relationsfor is a term encountered in some mathematical and computer science discussions to denote the operation of extracting, for a given element, the set of related elements via a relation. It is not a standard name in formal textbooks, but is used informally as a reminder of the direction of the relation from inputs to outputs.

Formally, let R be a relation between sets A and B, i.e., R ⊆ A × B. For a ∈ A, relationsfor(R, a) is the subset of B defined by { b ∈ B | (a, b) ∈ R }. If R is a function, this set is either empty or a singleton {f(a)}; in a general relation it may contain multiple elements.

Example: Let A = {1, 2}, B = {x, y}, and R = { (1, x), (1, y), (2, y) }. Then relationsfor(R, 1) = {x, y}. If R represents a function f: {1,2} → {x,y} with f(1) = x, f(2) = y, then relationsfor(R, 1) = {x}.

Usage appears in discussions of databases, graphs, and logic as a way to describe the related-neighborhood of an element under a relation. It is closely related to the relational image or forward image of a relation, and to the concept of adjacency in directed graphs. In databases, evaluating relationsfor can resemble performing a join on a key to obtain related records.

As a formal term, the standard terminology is relational image or forward image; relationsfor is a descriptive label used in some teaching materials and code libraries. See also: relation, relation image, functional relation, graph, database join.