Set of functions from empty set to $\{0,1\}$ How does the set of all functions $\{f \,|\, f: \emptyset \to \{0,1\}\}$ look like? Is it empty or does it contain infinitely many functions? Does the definition $f: \emptyset \to \{0,1\}$ make sense at all?
I was wondering because we know that the two sets $\{0,1\}^X$ and $\mathcal{P}(X)$ have the same cardinality. But this is only true if $X$ is non-empty, right?

It is known as the empty function.
For any set $A$, there is exactly one function from the empty set to $A$, namely the empty function:
$$f_A: \emptyset \to A.$$
The graph of an empty function is a subset of the Cartesian product $\emptyset × A$. Since the product is empty the only such subset is the empty set $\emptyset$.
