Definition:Relation/Relation as Subset of Cartesian Product

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Definition

Most treatments of set theory and relation theory define a relation on $S \times T$ to refer to just the truth set itself:

$\RR \subseteq S \times T$

where:

$S \times T$ is the Cartesian product of $S$ and $T$.


Thus under this treatment, $\RR$ is a set of ordered pairs, the first coordinate from $S$ and the second coordinate from $T$.

This approach leaves the precise nature of $S$ and $T$ undefined.


Also defined as

As a sideline, it is noted that some sources define a relation $\RR$ as a set of ordered pairs, with no initial reference to the domain or image of $\RR$.

The domain and image of $\RR$ are then defined as the sets:

\(\ds \Dom \RR\) \(=\) \(\ds \set {x: \exists y: \tuple {x, y} \in \RR}\)
\(\ds \Img \RR\) \(=\) \(\ds \set {y: \exists x: \tuple {x, y} \in \RR}\)

Using this approach, the cartesian product $S \times T$ of two sets $S$ and $T$ is defined as the relation consisting of all the ordered pairs $\tuple {x, y}$ where $x \in S$ and $y \in T$, rather than defining the cartesian product first and the relation as being a subset of it.


Also see


Sources

where the remark is made with reference to the definition of a mapping
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