Definition:Inverse Image

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 $$D \subseteq T$$.

The inverse image of $$D$$ under $$\mathcal{R}$$ is the set:
 * $$\left\{{s \in S: \mathcal{R} \left({s}\right) \in D}\right\}$$

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