Domain of Relation is Image of Inverse Relation

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

Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse of $\mathcal R$.


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

That is, the domain of a relation is the image of its inverse.


By definition:

$\operatorname{Dom} \left({\mathcal R}\right) := \left\{{s \in S: \exists t \in T: \left({s, t}\right) \in \mathcal R}\right\}$
$\operatorname{Im} \left({\mathcal R^{-1}}\right) := \left\{{s \in S: \exists T \in T: \left({t, s}\right) \in \mathcal R^{-1}}\right\}$

\(\displaystyle x\) \(\in\) \(\displaystyle \operatorname{Dom} \left({\mathcal R}\right)\)
\(\displaystyle \iff \ \ \) \(\displaystyle \exists t \in T: \left({x, t}\right)\) \(\in\) \(\displaystyle \mathcal R\) Definition of Domain of Relation
\(\displaystyle \iff \ \ \) \(\displaystyle \exists t \in T: \left({t, x}\right)\) \(\in\) \(\displaystyle \mathcal R^{-1}\) Definition of Definition:Inverse Relation
\(\displaystyle \iff \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \operatorname{Im} \left({\mathcal R^{-1} }\right)\) Definition of Image of Relation


Also see