Partial function

A partial function of the amount of X in the amount of Y is a rather unique relation, that is, a relation in which each element of the set X, a maximum element of the set Y is assigned. The notion of partial function is used in theoretical computer science, especially in computability theory.

The term of the partial function is a generalization of the concept of the function. Among a function from X to Y is defined as a left total, rather unequivocal relation, ie a relation that is in every element of X assigned to exactly one element of Y. Each function of X to Y is therefore in particular a partial function from X to Y, but not vice versa. In this respect, the notion of partial function is misleading. To express that a partial function is a function, it is said occasionally, there were a total function. The difference between the partial and total features is that applies for partial functions and total features.

As a domain of the partial function is defined as the set of all the elements of X, to which is assigned an element of Y. A partial function f is exactly then a function when applies.

A partial function f from X to Y can be modeled in two different ways as a function:

The value ( "bottom" or "undefined" ) may not be to Y.

Spellings

For " f is a partial function from X to Y" to write: or or or as well. Not recommended is the notation, as it defines f as a function, which is not well defined in general.

The notation " is undefined " or even " " is problematic, because the expression is precisely not allowed then. Clearly it is to say "f is undefined at x " or as formula " ".

Examples

  • The partial function is undefined at the point, because division by 0 is not allowed in the real numbers. Can be formed
  • Partial - recursive functions
  • An unbounded linear operator

Applications

If an algorithm takes inputs from the set X, and provides outputs of the set Y, then it calculates a partial function from X to Y. The domain of this function is the set of all elements of X, to which the algorithm provides a value. In order to provide a value, it must, in particular with its calculation come to an end ( terminate ).

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