Definition:Relation/Relation as Mapping
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Definition
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 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts.
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 3$: Equivalence relations