The sentence “Alice is taller than Bob” describes a relation between two people. So does “3 is less than 7,” or “Topeka is the capital of Kansas.” In each case, there is a pair of things and a rule that says whether the pair is connected. In mathematics, a relation is any such rule: given two objects, the relation either holds between them or it does not.
More precisely, a binary relation on a set is a collection of ordered pairs where both and belong to . The pair is in the relation exactly when the relation holds from to . Writing means “the relation holds between and .” The “less than” relation on whole numbers contains the pair but not the pair , because is true and is false.
Three properties distinguish different kinds of relations:
- Reflexive: every element is related to itself. Equality is reflexive, because for anything.
- Symmetric: if , then . “Is a sibling of” is symmetric. “Is taller than” is not.
- Transitive: if and , then . “Is an ancestor of” is transitive.
A relation that is reflexive, symmetric, and transitive is called an equivalence relation and partitions its set into groups of mutually related elements. A relation that is reflexive, antisymmetric (if and then ), and transitive is called a partial order and arranges its elements into a hierarchy.
Relations matter throughout this vault because the semiotic universe is built on relations among signs. The broader project of relationality treats relations as ontologically prior to the things they connect: objects are constituted through their relations rather than existing independently. Mathematical relations give this philosophical stance its formal backbone.
Related concepts
- Set theory — the framework in which relations are usually defined
- Relationality — the philosophical framework grounding this vault
- Function — a special kind of relation where each input has exactly one output
- Partial order — a relation that is reflexive, antisymmetric, and transitive
- Semiotic universe — the formal structure built on sign relations