# 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also known as

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

The term preimage set is also seen.