Definition:Inverse Relation

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

The inverse 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:


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

Also known as
An inverse relation is also seen as converse relation.

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

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

Others use $\breve {\mathcal R}$, for example and.

Also see

 * Image of Relation is Domain of Inverse Relation
 * Domain of Relation is Image of Inverse Relation


 * Definition:Inverse of Mapping
 * Definition:Inverse Mapping


 * Definition:Dual Ordering: the inverse of an ordering.