# Cardinality of Set of All Mappings from Empty Set

## Theorem

Let $T$ be a set.

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

Then:

$\left\vert{T^\varnothing}\right\vert = 1$

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

## Proof

The only element of $T^\varnothing$ is the null relation:

$\varnothing \times T$

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

So:

$\left\vert{T^\varnothing}\right\vert = 1$

$\blacksquare$