Empty function

In mathematics, an empty function is a function whose domain is the empty set $\emptyset$ . For each set $A$ , there is exactly one such empty function

f_{A}:\varnothing \rightarrow A.

The graph of an empty function is a subset of the Cartesian product $\emptyset \times A$ . Since the product is empty the only such subset is the empty set $\emptyset$ . The empty subset is a valid graph since for every $x$ in the domain $\emptyset$ there is a unique $y$ in the codomain $A$ such that $(x, y) \in \emptyset \times A$ . This statement is an example of a vacuous truth since "there is no $x$ in the domain."

The existence of an empty function from $\emptyset$ to $\emptyset$ is required to make the category of sets a category, because in a category, each object needs to have an "identity morphism", and only the empty function is the identity on the object $\emptyset$ . The existence of a unique empty function from $\emptyset$ into each set $A$ means that the empty set is an initial object in the category of sets. In terms of cardinal arithmetic, it means that $k 0 = 1$ for every cardinal number $k$ —particularly profound when $k = 0$ to illustrate the strong statement of indices pertaining to $0$ .

References

Herrlich, Horst and Strecker, George E., Category Theory, Heldermann Verlag (2007).