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. 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 ∅×A. Since the product is empty the only such subset is the empty set ∅. The empty subset is a valid graph since for every x in the domain ∅ there is a unique y in the codomain A such that (x,y) ∈ ∅. This is an example of a vacuous truth since there is no x in the domain.

Most authors will not care, when defining the term “constant function” precisely, 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 of not considering 1 to be a prime number, an empty topological space to be connected, or the trivial group to be simple.

The existence of an empty function from ∅ to ∅ is required to make the category of sets a category. (In a category, each object needs to have an "identity morphism", and only the empty function is the identity on the object ∅.) The existence of a unique empty function from ∅ 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 k⁰ = 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, Allen and Bacon, Inc. Boston (1973).