World Library  
Flag as Inappropriate
Email this Article

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) ∈ ∅ × 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 ∅ to ∅ 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 ∅. 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.

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 of not considering 1 to be a prime number, an empty topological space to be connected, or the trivial group to be simple.


  • Herrlich, Horst and Strecker, George E., Category Theory, Heldermann Verlag (2007).
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.