Definition:Relation

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

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

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

Not all elements in $S$ need to be related to every element in $T$ (but see Trivial Relation).

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

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: $s \not\mathrel{\mathcal R} t$, that is, by drawing a line through the relation symbol.

See Complement of Relation.

Also known as
In the field of predicate logic, a relation can be seen referred to as a relational property.

Some sources, for example, use the term correspondence for what is defined here as relation, reserving the term relation for what on is defined as endorelation, that is, a relation on $S \times S$ for some set $S$.

As this can cause confusion with the usage of correspondence to mean a relation which is both left-total and right-total, it is recommended that this is not used.

Some sources prefer the term relation between $S$ and $T$ as it can be argued that this provides better emphasis on the existence of the domain and codomain.

refers to a correspondence between $S$ and $T$.

Also see

 * Definition:Endorelation
 * Definition:Correspondence


 * Definition:Mapping


 * Definition:Relational Structure


 * Definition:Domain of Relation
 * Definition:Range


 * Definition:Entourage


 * Definition:Complement of Relation


 * Definition:Characteristic Function/Relation


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

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

Technical Note
The expression:


 * $s \mathrel {\mathcal R} t$

is produced by the following $\LaTeX$ code:

s \mathrel {\mathcal R} t