relations

Relations describe connections between entities. In mathematics, a binary relation on a set A is a subset of the Cartesian product A × A, consisting of ordered pairs (a, b) where a is related to b. More broadly, relations can connect objects of different types, and in data modeling a relation is often represented as a table that stores linked items.

Relations can have formal properties. A relation is reflexive if every element relates to itself; symmetric if a relates to b implies b relates to a; antisymmetric if a relates to b and b relates to a imply a = b; and transitive if a relates to b and b relates to c implies a relates to c. An equivalence relation is reflexive, symmetric, and transitive; a partial order is reflexive, antisymmetric, and transitive. A total order also requires comparability of all elements.

Examples include equality on numbers, the divisibility relation a divides b, and congruence modulo n. A relation can be drawn as a directed graph with nodes for elements and edges for related pairs; composition of relations combines them, and the inverse relation swaps the order of pairs.

Applications span mathematics, logic, and computer science. In databases the relational model represents data as relations (tables) and uses relational algebra to query it. In logic, relations are predicates that express how objects stand in relation to one another; in social sciences, “relation” often refers to interpersonal ties such as family, friendship, or professional connections.