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.

Note
It is not formally correct to consider a relation as consisting of merely the set $\mathcal R$, as this renders the concept of codomain ill-defined.

Formally, when considering a relation as the ordered pair $\left({S \times T, \mathcal R}\right)$, the concept of codomain is, in general, still ill-defined for the same reason. In the case that $S$ is empty, it follows that $S \times T$ is empty, but $T$ is not uniquely determined.

The interpretation of a relation as the ordered triple $\left({S, T, \mathcal R}\right)$ removes this issue.

In can also be noted that the term "relation on $S \times T$" does not, in general, specify either the domain or the codomain of the relation.

On the other hand, the alternative term "relation between $S$ and $T$" does. In addition, it removes the potential ambiguity of the phrase "relation on $S \times T$", which can refer to either a subset of $S \times T$ or a subset of $\left({S \times T}\right) \times \left({S \times T}\right)$, as the word "relation" is often substituted for "endorelation". Arguably, the former use renders the term "relation" superfluous, as (in that case) the phrase "relation on" is (by definition) synonymous with "subset of".

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.

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