Image of Empty Set is Empty Set
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Theorem
Let $\RR \subseteq S \times T$ be a relation.
The image of the empty set is the empty set:
- $\RR \sqbrk \O = \O$
Corollary 1
Let $f: S \to T$ be a mapping.
The image of the empty set is the empty set:
- $f \sqbrk \O = \O$
Corollary 2
Let $S = \O$ and $T \ne \O$.
There is no surjection $f: S \to T$.
Proof
\(\ds \RR \sqbrk \O\) | \(=\) | \(\ds \set {t \in \Rng \RR: \exists s \in \O: \tuple {s, t} \in \RR}\) | Definition of Image of Subset under Relation | |||||||||||
\(\ds \neg \exists s\) | \(\in\) | \(\ds \O\) | Definition of Empty Set | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \neg \exists t\) | \(\in\) | \(\ds \set {t \in \Rng \RR: \exists s \in \O: \tuple {s, t} \in \RR}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \RR \sqbrk \O\) | \(=\) | \(\ds \O\) | Definition of Empty Set |
$\blacksquare$