partial function

partial function (plural partial functions )

1. (mathematics ) A function whose domain is a subset of the set on which it is formally defined; i.e., a function f: X→Y for which values f(x) are defined only for x ∈ W, where W ⊆ X.
   • 1967 [John Wiley & Sons], Stephen Cole Kleene, Mathematical Logic, 2002, Dover, page 244,

     The Church-Turing thesis applies to partial functions on the same grounds as to total functions (§ 41).

   • 1991, Michel Bidoit, Hans-Jörg Kreowski, Pierre Lescanne, Fernando Orejas, Donald Sannella (editors), Algebraic System Specification and Development: A Survey and Annotated Bibliography, Springer, LNCS 501, page 15,

     Nowadays it seems quite clear that if algebraic specifications are to be used as a powerful and realistic tool for the development of complex systems they should permit the specification of partial functions. [...] There are essentially two ways of specifying partial functions.

   • 2006, Paulo Oliva, Understanding and Using Spector's Bar Recursive Interpretation of Classical Analysis, Arnold Beckmann, Ulrich Berger, Benedikt Löwe, John V. Tucker (editors), Logical Approaches to Computational Barriers: 2nd Conference on Computability in Europe, Proceedings, Springer, LNCS 3988, page 432,

     In the following ${\displaystyle \sigma }${\displaystyle \sigma } and ${\displaystyle \tau }${\displaystyle \tau } will denote finite partial functions from ${\displaystyle \mathbb {N} }${\displaystyle \mathbb {N} } to ${\displaystyle \mathbb {N} }${\displaystyle \mathbb {N} }, i.e. partial functions which are defined on a finite domain. A partial function which is everywhere undefined is denoted by ${\displaystyle \langle \ \rangle }${\displaystyle \langle \ \rangle }, whereas a partial function defined only at position ${\displaystyle k}${\displaystyle k} (with value ${\displaystyle n}${\displaystyle n}) is denoted by ${\displaystyle \langle k,n\rangle }${\displaystyle \langle k,n\rangle }.

This is not a formal term, but a metamathematical description which only assumes concrete meaning in context. For example, in computability theory, a partial function is a function whose domain is a subset of ${\displaystyle {\mathbb {N} }^{k}}${\displaystyle {\mathbb {N} }^{k}} for some k, but in other fields the term has other meanings.

• (antonym(s) of "mathematics: function whose domain is a subset of the set on which it is formally defined"): total function

• Injective function on Wikipedia.Wikipedia
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