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: