Partial function

An example of a partial function that is not a total function.

An example of a partial function that is also a total function.

In mathematics, a partial function from X to Y is a function f: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y (only some subset ). If X' = X, then f is called a total function and is equivalent to a function. Partial functions are often used when the exact domain, X' , is not known (e.g. many functions in computability theory).

Specifically, we will say that for any , either:

• (it is defined as a single element in Y) or
• f(x) is undefined.

For example we can consider the square root function restricted to the integers

Thus g(n) is only defined for n which are perfect squares (i.e. 0, 1, 4, 9, 16, ...). So, g(25) = 5, but g(26) is undefined.

Domain of a partial function

There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function. Most mathematicians, including recursion theorists, use the term "domain of f" for the set of all values x such that f(x) is defined ( X' above). But some, particularly category theorists, consider the domain of a partial function f:X→Y to be X, and refer to X' as the domain of definition.

Discussion and examples

The first diagram above represents a partial function that is not a total function since the element 1 in the left-hand set is not associated with anything in the right-hand set.

Natural logarithm

Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore, the natural logarithm function is not a total function when viewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only include the positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals), then the natural logarithm is a total function.

Subtraction of natural numbers

Subtraction of natural numbers (non-negative integers) can be viewed as a partial function:

f(x,y) = x − y

It is only defined when .

Bottom type

In some automated theorem proving systems a partial function is considered as returning the bottom type when it is undefined. The Curry-Howard correspondence uses this to map proofs and computer programs to each other.

In computer science a partial function corresponds to a subroutine that raises an exception or loops for ever. The IEEE floating point standard defines a Not-a-number value which is returned when a floating point operation is undefined and exceptions are suppressed, e.g. when the square root of a negative number is requested.

See also

• Bijection
• Injective function
• Surjective function
• Multivalued function
• Symmetric inverse semigroup

References

• Martin Davis (1958), Computability and Unsolvability, McGraw-Hill Book Company, Inc, New York. Republished by Dover in 1982. ISBN 0-486-61471-9.
• Stephen Kleene (1952), Introduction to Meta-Mathematics, North-Holland Publishing Company, Amsterdam, Netherlands, 10th printing with corrections added on 7th printing (1974). ISBN 0-7204-2103-9.
• Harold S. Stone (1972), Introduction to Computer Organization and Data Structures, McGraw-Hill Book Company, New York.