Partial function
From Exampleproblems
This article pertains to functions in mathematics and computer science. For other usages see function (disambiguation).
In mathematics, a partial function is a relation, such that each element of a set (the domain) is associated with at most one element of another (possibly the same) set (the codomain). This means that some elements of the domain may not be associated to any element of the codomain.
If a partial function associates every element in its domain with precisely one element of its codomain, then it is termed a total function, or simply a "function" as traditionally understood in mathematics. Not every partial function is a total function.
Discussion and examples
The above diagram represents a partial function that is not a total function since the element 1 in X is not associated with anything. Until the second half of the 20th century, only total functions were considered "well-defined".
The natural logarithm function from the real numbers to the reals is only partial, as the logarithm of non-positive reals is not a real number.
