Relaatiot

Relaatiot, literally “relations” in Finnish, denote connections between elements in one or more sets. In mathematics and logic, a relation between sets A1, ..., Ak is a subset of their Cartesian product A1 × A2 × ... × Ak. A binary relation, on a single set A, is a subset R ⊆ A × A and consists of those ordered pairs (a, b) that satisfy the relation.

Common examples include equality, order, and adjacency relations. The relation “a is related to b” may be reflexive (every element relates to itself), symmetric (if a relates to b, then b relates to a), antisymmetric (if a relates to b and b relates to a then a = b), or transitive (if a relates to b and b relates to c then a relates to c). These properties lead to important special classes: equivalence relations (reflexive, symmetric, transitive) partition a set into equivalence classes; partial orders (reflexive, antisymmetric, transitive) model hierarchical structures; linear orders are total orders.

Relations can be represented as directed graphs, where vertices are elements and edges indicate related pairs, or as matrices in which an entry is 1 if the pair is related. In computer science, the relational model uses relations (tables) to store data; relational algebra and SQL manipulate such relations. In linguistics, semantic and syntactic relations connect words and structures. Higher-arity relations connect more than two elements.