Empty 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 are not any 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 a unique empty function for each set A means that the empty set is an initial object in the category of sets.

References

• Herrlich, Horst and Strecker, George E.; Category Theory, Allen and Bacon, Inc. Boston (1973).

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