\begin{definition}[Definition:Relation/Relation as Mapping]
It is possible to define a relation as a mapping from the cartesian product $S \times T$ to the set of truth values $\set {\text {true}, \text {false} }$:
:$\RR: S \times T \to \set {\text {true}, \text {false} }: \forall \tuple {s, t} \in S \times T: \map \RR {s, t} = \begin{cases}
\text {true} & : \tuple {s, t} \in \RR \\
\text {false} & : \tuple {s, t} \notin \RR
\end{cases}$
This is called the characteristic function of the relation.
However, care needs to be taken that a mapping then cannot be defined as a special relation, as this would be circular.
This approach is taken in {{BookReference|Lectures in Abstract Algebra|1951|Nathan Jacobson|volume = I|subtitle = Basic Concepts}}.
\end{definition}