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:


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

That is, the preimage of $Y$ under $\mathcal R$ is the image of $Y$ under $\mathcal R^{-1}$.

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

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

The term preimage set is also seen.

As well as using the notation $\Preimg {\mathcal R}$ to denote the preimage of an entire relation, the symbol $\operatorname {Img}^{-1}$ can also be used as follows:

For $Y \subseteq \Img {\mathcal R}$:
 * $\map {\operatorname {Img}^{-1}_\mathcal R} Y := \mathcal R^{-1} \sqbrk Y$

but this notation is rarely seen.

Some authors use $\map {\mathcal R^\gets} Y$ for what we have here as $\mathcal R^{-1} \sqbrk Y$.

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