Definition:Relation

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

A relation (in this context, technically speaking, a binary relation) on $S \times T$ is an arbitrary subset $\mathcal R \subseteq S \times T$.

What this means is that a binary relation relates (certain) elements of one set with (certain) elements of another.

Not all elements in $S$ need to be related to every relation in $T$ (but see Trivial Relation, which is a relation in which they are).

When $\left({s, t}\right) \in \mathcal R$, we can write:
 * $s \mathop {\mathcal R} t$

or:
 * $\mathcal R \left({s, t}\right)$.

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

If $\left({s, t}\right) \notin \mathcal R$, we can write: $s \not \mathcal R t$, that is, by drawing a line through the relation symbol. See Complement of Relation.

Also see

 * Mapping


 * Relational Structure


 * Domain
 * Range


 * Entourage


 * Complement of Relation


 * Characteristic Function of a Relation

Linguistic Note
In natural language what we have defined as a relation is usually understood as a relationship.