Definition:Preimage/Relation/Relation

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

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


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

The preimage of $\mathcal R \subseteq S \times T$ is:


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

Preimage of Subset as Element of Inverse Image Mapping
The preimage of $Y$ under $f$ can be seen to be an element of the codomain of the inverse image mapping $\mathcal R^\gets: \powerset T \to \powerset S$ of $\mathcal R$:


 * $\forall Y \in T: \map {\mathcal R^\gets} Y := \set {s \in S: \exists y \in Y: \tuple {s, y} \in \mathcal R}$

Both approaches to this concept are used in

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

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


 * Definition:Image of Relation