functions from empty set

Sometimes, it is useful to consider functions whose domain is theempty set. Given a set, there exists exactly one function fromthe the empty set to that set. The rationale for this comes fromcarefully examining the definition of function in this degeneratecase. Recall that, in set theory, a function from a set $D$ to aset $R$ is a set of ordered pairs whose first element lies in $D$and whose second element lies in $R$ such that every element of $D$appears as the first element of exactly one ordered pair. If wetake $D$ to be the empty set, we see that this definition is satisfiedif we take our function to be set of no ordered pairs — since thereare no elements in the empty set, it is technically correct to saythat every element of the empty set appears as a first element ofan ordered pair which is an element of the empty set!

This observation turns out to be more than just an exercise inlogic, being useful in several contexts. Given a set $S$ and a positiveinteger $n$, we may define $S^n$ as the set of all functions from$\{1,\mathrm{\dots },n\}$ to $S$. If we choose $n=0$, then $S^0$ consistsof all maps from the empty set to $S$, hence consists of exactly oneelement — see the entry on empty products for a discussion of theusefulness of this convention. In category theory, it turns out thatfunctions from the empty set are important because they make the emptyset be an initial object in this category.