Definition:Inverse Relation

Definition
Let $\mathcal R \subseteq S \times T$ be a relation.

The inverse (or converse) relation to (or of) $\mathcal R$ is defined as:


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

That is, $\mathcal R^{-1} \subseteq T \times S$ is the relation which satisfies:


 * $\forall s \in S: \forall t \in T: \left({t, s}\right) \in \mathcal R^{-1} \iff \left({s, t}\right) \in \mathcal R$

Domain and range of Inverse Relation
Note that the preimage of a relation is the image of its inverse, and vice versa:


 * $\operatorname{Im}^{-1} \left({\mathcal R}\right) = \operatorname{Im} \left({\mathcal R^{-1}}\right)$
 * $\operatorname{Im} \left({\mathcal R}\right) = \operatorname{Im}^{-1} \left({\mathcal R^{-1}}\right)$

Alternative Notations
Some authors use the notation $\mathcal R^\gets$ instead of $\mathcal R^{-1}$.

Others, for example, use $\mathcal R^t$.

And others, e.g., use $\breve{\mathcal R}$.

Also see

 * Inverse of a Mapping
 * Inverse Mapping