Empty function

Not to be confused with identity function or null 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 $0$ .

When defining the term "constant function" precisely, most authors will not care whether or not the empty function qualifies, and will use whatever definition is most convenient. Sometimes, however, it is best not to consider the empty function to be constant, and a definition that makes reference to the range is preferable in those situations. This is much along the same lines as not considering 1 to be a prime number (1 is now considered the empty product), an empty topological space to be connected, or the trivial group to be simple.

References

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

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