# functions from empty set

Sometimes, it is useful to consider functions whose domain is the
empty set^{}. Given a set, there exists exactly one function from
the the empty set to that set. The rationale for this comes from
carefully examining the definition of function in this degenerate
case. Recall that, in set theory^{}, a function from a set $D$ to a
set $R$ 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 we
take $D$ to be the empty set, we see that this definition is satisfied
if we take our function to be set of no ordered pairs — since there
are no elements in the empty set, it is technically correct to say
that every element of the empty set appears as a first element of
an ordered pair which is an element of the empty set!

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

Title | functions from empty set |
---|---|

Canonical name | FunctionsFromEmptySet |

Date of creation | 2013-03-22 18:08:20 |

Last modified on | 2013-03-22 18:08:20 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 5 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 03-00 |