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:

$\card {T^\O} = 1$

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 8$: Functions: Exercise $\text{(i)}$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory: Exercise $1.9 \ \text{(b)}$