In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition of f. If S equals X, that is, if f is defined on every element in X, then f is said to be total.
More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to exactly one element of the second set.
A partial function is often used when its exact domain of definition is not known or difficult to specify. This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a function. In computability theory, a general recursive function is a partial function from the integers to the integers; for many of them no algorithm can exist for deciding whether they are in fact total.
When arrow notation is used for functions, a partial function from to is sometimes written as f: X ⇸ Y, or However, there is no general convention, and the latter notation is more commonly used for injective functions..
Specifically, for a partial function and any one has either:
- (it is a single element in Y), or
- is undefined.
- defined by:
- if, and only if,
then is only defined if is a perfect square (that is, ). So but is undefined.
A partial function is said to be injective, surjective, or bijective when the function given by the restriction of the partial function to its domain of definition is injective, surjective, bijective respectively.
An injective partial function may be inverted to an injective partial function, and a partial function which is both injective and surjective has an injective function as inverse. Furthermore, a function which is injective may be inverted to an injective partial function.
A function is a binary relation that is functional (also called right-unique) and serial (also called left-total). This is a stronger definition than that of a partial function which only requires the functional property.
The set of all partial functions from a set to a set denoted by is the union of all functions defined on subsets of with same codomain :
the latter also written as In finite case, its cardinality is
because any partial function can be extended to a function by any fixed value not contained in so that the codomain is an operation which is injective (unique and invertible by restriction).
Discussion and examples
The first diagram at the top of the article represents a partial function that is not a function since the element 1 in the left-hand set is not associated with anything in the right-hand set. Whereas, the second diagram represents a function since every element on the left-hand set is associated with exactly one element in the right hand set.
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 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 function.
Subtraction of natural numbers
It is defined only when
In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. 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.
In a programming language where function parameters are statically typed, a function may be defined as a partial function because the language's type system cannot express the exact domain of the function, so the programmer instead gives it the smallest domain which is expressible as a type and contains the domain of definition of the function.
In category theory
In category theory, when considering the operation of morphism composition in concrete categories, the composition operation is a function if and only if has one element. The reason for this is that two morphisms and can only be composed as if that is, the codomain of must equal the domain of
The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps. One textbook notes that "This formal completion of sets and partial maps by adding “improper,” “infinite” elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science."
In abstract algebra
Partial algebra generalizes the notion of universal algebra to partial operations. An example would be a field, in which the multiplicative inversion is the only proper partial operation (because division by zero is not defined).
The set of all partial functions (partial transformations) on a given base set, forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on ), typically denoted by  The set of all partial bijections on forms the symmetric inverse semigroup.
Charts and atlases for manifolds and fiber bundles
Charts in the atlases which specify the structure of manifolds and fiber bundles are partial functions. In the case of manifolds, the domain is the point set of the manifold. In the case of fiber bundles, the domain is the space of the fiber bundle. In these applications, the most important construction is the transition map, which is the composite of one chart with the inverse of another. The initial classification of manifolds and fiber bundles is largely expressed in terms of constraints on these transition maps.
The reason for the use of partial functions instead of functions is to permit general global topologies to be represented by stitching together local patches to describe the global structure. The "patches" are the domains where the charts are defined.
- Analytic continuation – Extension of the domain of an analytic function (mathematics)
- Multivalued function – Generalization of a function that may produce several outputs for each input
- Densely defined operator – Function that is defined almost everywhere (mathematics)
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