Empty Mapping is Surjective iff Codomain is Empty

Theorem

Let $T$ be a set.

Let $\O$ denote the empty set.


Let $e: \O \to T$ be the empty mapping.

Then $e$ is a surjection if and only if $T = \O$.


Proof

Let $T = \O$.

From Empty Mapping to Empty Set is Bijective, $e$ is a bijection.

Hence a fortiori $e$ is a surjection.

$\Box$


Let $T \ne \O$.

Aiming for a contradiction, suppose $e$ is a surjection.

Let $t \in T$.

As $e$ is a surjection:

$(1): \quad \exists s \in S: \map e s = t$

But by Null Relation is Mapping iff Domain is Empty Set:

$\Dom e = \O$

Hence:

$\nexists s \in S: \map e s = t$

This is a contradiction of $(1)$.

Hence by Proof by Contradiction it is not possible for $e$ to be a surjection.

Hence if $e$ is a surjection it must be the case that $T = \O$.

$\Box$


So we have:

$T = \O$ implies that $e$ is a surjection
$e$ is a surjection implies that $T = \O$

and the result follows.

$\blacksquare$

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