Image of Empty Set is Empty Set

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Theorem

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


The image of the empty set is the empty set:

$\mathcal R \sqbrk \O = \O$


Corollary

Let $f: S \to T$ be a mapping.


The image of the empty set is the empty set:

$f \sqbrk \O = \O$


Proof

\(\displaystyle \mathcal R \sqbrk \O\) \(=\) \(\displaystyle \set {t \in \Rng {\mathcal R}: \exists s \in \O: \tuple {s, t} \in \mathcal R}\) Definition of Image of Subset under Relation
\(\displaystyle \neg \exists s\) \(\in\) \(\displaystyle \O\) Definition of Empty Set
\(\displaystyle \leadsto \ \ \) \(\displaystyle \neg \exists t\) \(\in\) \(\displaystyle \set {t \in \Rng {\mathcal R}: \exists s \in \O: \tuple {s, t} \in \mathcal R}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \mathcal R \sqbrk \O\) \(=\) \(\displaystyle \O\) Definition of Empty Set

$\blacksquare$