Image of Empty Set is Empty Set

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

$\blacksquare$