Definition:Relation/Relation as Ordered Pair

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

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings
• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations