Partial function: Difference between revisions

Revision as of 10:26, 24 September 2024

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More technically, a partial function is a [[binary relation]] over two [[Set (mathematics)|sets]] that associates to every element of the first set ''at most'' one element of the second set; it is thus a [[univalent relation]]. This generalizes the concept of a (total) [[Function (mathematics)|function]] by not requiring ''every'' element of the first set to be associated to an element of the second set.

A partial function is often used when its exact ___domain of definition is not known, or is difficult to specify. However, even when the exact ___domain of definition is known, partial functions are often used for simplicity or brevity. This is the case in [[mathematical analysiscalculus]], where, for example, the [[quotient]] of two functions is a partial function whose ___domain of definition cannot contain the [[Zero of a function|zeros]] of the denominator.; Forin this reasoncontext, a partial function is generally simply called a {{em|function}}.

In [[computability theory]], a [[general recursive function]] is a partial function from the integers to the integers; no [[algorithm]] can exist for deciding whether an arbitrary such function is in fact total.