Cardinality of Set of All Mappings to Empty Set

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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.

$\blacksquare$


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