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Helped to understand the definition of partial functions in terms of binary relation. Tags: Reverted Visual edit |
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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'''.
In other words, 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]]. (There may be some elements in the ___domain that are not mapped to elements in the ___domain.) 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 [[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; in this context, a partial function is generally simply called a {{em|function}}.
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