Definition:Relation

Definition
Let $S \times T$ be the cartesian product of two sets or classes $S$ and $T$.

A relation on $S \times T$ is an ordered triple:
 * $\mathcal R = \tuple {S, T, R}$

where $R \subseteq S \times T$ is a subset of the Cartesian product of $S$ and $T$.

What this means is that a relation relates (certain) elements of one set or class $S$ with (certain) elements of another, $T$.

Not all elements of $S$ need to be related to every (or even any) element of $T$ (but see Trivial Relation).

When $\tuple {s, t} \in R$, we can write:
 * $s \mathrel {\mathcal R} t$

or:
 * $\map {\mathcal R} {s, t}$

and can say $s$ bears $\mathcal R$ to $t$.

If $\tuple {s, t} \notin R$, we can write: $s \not \mathrel {\mathcal R} t$, that is, by drawing a line through the relation symbol.

See Complement of Relation.

Also see

 * Definition:Endorelation
 * Definition:Correspondence


 * Definition:Mapping


 * Definition:Relational Structure


 * Definition:Domain of Relation
 * Definition:Codomain of Relation
 * Definition:Range


 * Definition:Entourage


 * Definition:Complement of Relation


 * Definition:Characteristic Function of Relation


 * Definition:Trivial Relation, the relation on $S \times T$ in which every element of $S$ is related to every element of $T$.