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:


 * $\mathcal R^{-1} = \left\{{\left({t, s}\right): \left({s, t}\right) \in \mathcal R}\right\}$

Let $Y \subseteq T$.

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


 * $\mathcal R^{-1} \left ({Y}\right) := \left\{{s \in S: \exists y \in Y: \left({s, y}\right) \in \mathcal R}\right\}$

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

Clearly:
 * $\displaystyle \mathcal R^{-1} \left ({Y}\right) = \bigcup_{y \in Y} \mathcal R^{-1} \left({y}\right)$

... the union of the preimages of each of the elements of $Y$.

If no element of $Y$ has a preimage, then $\mathcal R^{-1} \left ({Y}\right) = \varnothing$.

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 $\operatorname{Im}^{-1} \left ({\mathcal R}\right)$ to denote the preimage of an entire relation, the symbol $\operatorname{Im}^{-1}$ can also be used as follows:

For $Y \subseteq \operatorname{Im} \left({\mathcal R}\right)$:
 * $\operatorname{Im}^{-1}_\mathcal R \left ({Y}\right) := \mathcal R^{-1} \left ({Y}\right)$

but this notation is clumsy and generally not preferred.