Preimage of Subset under Relation equals Union of Preimages of Elements

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Theorem

Let $S$ and $T$ be sets.

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

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


Let $Y \subseteq T$ be a subset of $T$.


Then:

$\displaystyle \mathcal R^{-1} \left[{Y}\right] = \bigcup_{y \mathop \in Y} \mathcal R^{-1} \left({y}\right)$

where:

$\mathcal R^{-1} \left[{Y}\right]$ is the preimage of the subset $Y$ under $\mathcal R$
$\mathcal R^{-1} \left({y}\right)$ is the preimage of the element $y$ under $\mathcal R$.


Proof

By definition, $\mathcal R^{-1} \subseteq T \times S$ is a relation on $T \times S$.

Thus Image of Subset under Relation equals Union of Images of Elements can be applied directly.

$\blacksquare$