Definition:Preimage/Relation/Subset

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:


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

Let $Y \subseteq T$.

The preimage of $Y$ under $\mathcal R$ is defined as:


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

That is, the preimage of $Y$ under $\mathcal R$ is the image of $Y$ under $\mathcal R^{-1}$:
 * $\mathcal R^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {t, s} \in \mathcal R^{-1} }$

If no element of $Y$ has a preimage, then $\mathcal R^{-1} \sqbrk Y = \O$.

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


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

Thus:
 * $\forall Y \subseteq T: \mathcal R^{-1} \sqbrk Y = \map {\mathcal R^\gets} Y$

and so the preimage of $Y$ under $\mathcal R$ is also seen referred to as the inverse image of $Y$ under $\mathcal R$.

Both approaches to this concept are used in.

Also known as
The preimage of $Y$ is also known as the inverse image of $Y$.

The term preimage set is also seen.

Also see

 * Preimage of Subset under Relation equals Union of Preimages of Elements
 * Definition:Inverse Image Mapping of Relation

Special Cases

 * Definition:Preimage of Subset under Mapping

Generalizations

 * Definition:Preimage of Relation

Related Concepts

 * Definition:Image of Subset under Relation
 * Definition:Image of Subset under Mapping