# Partial function

In

mathematics and

computer science, a

**partial function**, from the

domain *X* to the

codomain *Y* is a

binary relation, over

*X* and

*Y*, which is

*functional*, that is, associates with every element in

set *X* with, at most, one

element in set

*Y*. If a partial function associates with every element in its domain

*precisely one* element of its

codomain, then it is a "

total function". Note that with this terminology, not every partial function is a "true" function.

This above diagron does not represent a "well-defined" function; because, the element 1, in *X*, is associated with nothing.

Partial functions are often used in theoretical computer science: the behavior of a

Turing machine for instance can be described by a partial function relating its inputs to its outputs. This is not in general a total function since a Turing machine does not always produce an output for every input: it can run into an infinite loop. Even worse, it can run into an infinite loop for different inputs.

See also: