Cardinality of Set of All Mappings from Empty Set

Theorem
Let $T$ be a set.

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

Then:
 * $\card {T^\O} = 1$

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

Proof
The only element of $T^\O$ is the null relation:
 * $\O \times T$

From Null Relation is Mapping iff Domain is Empty Set, $\O \times T$ is a mapping from $\O$ to $T$.

The result follows from Empty Mapping is Unique.

That is:
 * $\card {T^\O} = 1$