Cardinality of Set of All Mappings to Empty Set

Theorem
Let $S$ be a set.

Let $\varnothing^S$ be the set of all mappings from $S$ to $\varnothing$.

Then:
 * $\left\vert{\varnothing^S}\right\vert = \begin{cases}

1 & : S = \varnothing \\ 0 & : S \ne \varnothing \end{cases}$

where $\left\vert{\varnothing^S}\right\vert$ denotes the cardinality of $\varnothing^S$.

Proof
From Null Relation is Mapping iff Domain is Empty Set, the null relation:
 * $\mathcal R = \varnothing \subseteq S \times T$

is not a mapping unless $S = \varnothing$.

So if $S \ne \varnothing$:
 * $\left\vert{\varnothing^S}\right\vert = 0$

If $S = \varnothing$:
 * $\left\vert{\varnothing^S}\right\vert = 1$

Hence the result.