Definition:Relation/Relation as Mapping

Definition

It is possible to define a relation as a mapping from the cartesian product $S \times T$ to the set of truth values $\left\{{\text{true}, \text{false}}\right\}$:

$\mathcal R: S \times T \to \left\{{\text{true}, \text{false}}\right\}: \forall \left({s, t}\right) \in S \times T: \mathcal R \left({s, t}\right) = \begin{cases} \text{true} & : \left({s, t}\right) \in \mathcal R \\ \text{false} & : \left({s, t}\right) \notin \mathcal R \end{cases}$

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 1951: Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts.