Definition:Preimage/Relation/Relation

Definition
Let $\RR \subseteq S \times T$ be a relation.

Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$, defined as:


 * $\RR^{-1} = \set {\tuple {t, s}: \tuple {s, t} \in \RR}$

The preimage of $\RR \subseteq S \times T$ is:


 * $\Preimg \RR := \RR^{-1} \sqbrk T = \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$

Also known as
Some sources, for example, call this the domain of $\RR$.

However, this term is discouraged, as it is also seen used to mean the entire set $S$, including elements of that set which have no images.

Also see

 * Definition:Preimage of Mapping


 * Definition:Domain of Relation
 * Definition:Codomain of Relation
 * Definition:Range of Relation


 * Definition:Image of Relation