Total Function

A Total Function is a partial function that is defined for all Elements in its Input Set.

• Example(s):
  • a Scalar Field Function.
  • an Ackermann Function.
  • …
• Counter-Example(s):
  • a Partial Function.
• See: Total Relation, Subset, Domain of a Function.

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References

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/partial_function#Total_function Retrieved:2014-4-21.
  • Total function is a synonym for function. The use of the prefix "total" is to suggest that it is a special case of a partial function. For example, when considering the operation of morphism composition in Concrete Categories, the composition operation [math]\displaystyle{ \circ : \operatorname{Hom}(C) \times \operatorname{Hom}(C) \to \operatorname{Hom}(C) }[/math] is a total function if and only if [math]\displaystyle{ \operatorname{Ob}(C) }[/math] has one element. The reason for this is that two morphisms [math]\displaystyle{ f:X\to Y }[/math] and [math]\displaystyle{ g:U\to V }[/math] can only be composed as [math]\displaystyle{ g \circ f }[/math] if [math]\displaystyle{ Y=U }[/math], that is, the codomain of [math]\displaystyle{ f }[/math] must equal the domain of [math]\displaystyle{ g }[/math].

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