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{{more footnotes|date=August 2014}}
In [[mathematics]], a '''partial function''' {{mvar|f}} from a [[Set (mathematics)|set]] {{mvar|X}} to a set {{mvar|Y}} is a [[function (mathematics)||]] 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''' 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 partial 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 [[Binary relation#Special types of binary relations|functional binary relation]]. It generalizes the concept of a (total) [[Function (mathematics)|function]] by not requiring every element of the first set to be associated to ''exactly'' one element of the second set.
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