Partial function: Difference between revisions

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In [[mathematics]], a '''partial function''' {{mvar|f}} from a [[Set (mathematics)|set]] {{mvar|X}} to a set {{mvar|Y}} is a [[function (mathematics)|function]] from a [[subset]] {{mvar|S}} of {{mvar|X}} (possibly the whole {{mvar|X}} itself) to {{mvar|Y}}. The subset {{mvar|S}}, that is, the ''[[Domain of a function|___domain]]'' of {{mvar|f}} viewed as a function, is called the '''___domain of definition''' or '''natural ___domain''' of {{mvar|f}}. If {{mvar|S}} equals {{mvar|X}}, that is, if {{mvar|f}} is defined on every element in {{mvar|X}}, then {{mvar|f}} is said to be a '''total function'''.

More technically, a partialtotal function is a [[binary relation]] over two [[Set (mathematics)|sets]] that associates every element of the first set to ''at most'' one element of the second set; it is thus a [[univalent relation]]. ItPartial generalizesfunctions are a generalization of the concept of a (total) [[Function (mathematics)|function]] byvia 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 difficult to specify. This is the case in [[calculus]], 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. For this reason, in calculus, and more generally in [[mathematical analysis]], a partial function is generally called simply 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.