Definition:Relation/Relation as Ordered Pair

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Definition

Some sources define a relation between $S$ and $T$ as an ordered pair:

$\struct {S \times T, \map P {s, t} }$

where:

$S \times T$ is the Cartesian product of $S$ and $T$
$\map P {s, t}$ is a propositional function on ordered pairs $\tuple {s, t}$ of $S \times T$.


Note that this approach leaves the domain and codomain inadequately defined.

This situation arises in the case that $S$ or $T$ are empty, whence it follows that $S \times T$ is empty, but $T$ or $S$ are not themselves uniquely determined.


Sources