Cardinality of Set of All Mappings to Empty Set

Theorem
Let $S$ be a set.

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

Then:
 * $\card {\O^S} = \begin{cases}

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

where $\card {\O^S}$ denotes the cardinality of $\O^S$.

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

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

So if $S \ne \O$:
 * $\card {\O^S} = 0$

If $S = \O$:
 * $\card {\O^S} = 1$

Hence the result.