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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a partial function from a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
to a set is a function from a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of (possibly itself) to . The subset , that is, the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
of viewed as a function, is called the domain of definition of . If equals , that is, if is defined on every element in , then is said to be total. More technically, a partial function is a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, where, for example, the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
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 Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, a partial function is generally called simply a . In
computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since e ...
, a
general recursive function In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as i ...
is a partial function from the integers to the integers; no
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
can exist for deciding whether an arbitrary such function is in fact total. When arrow notation is used for functions, a partial function f from X to Y is sometimes written as f : X \rightharpoonup Y, f : X \nrightarrow Y, or f : X \hookrightarrow Y. However, there is no general convention, and the latter notation is more commonly used for
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota ...
s or
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
s. Specifically, for a partial function f : X \rightharpoonup Y, and any x \in X, one has either: * f(x) = y \in Y (it is a single element in ), or * f(x) is undefined. For example, if f is the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
function restricted to the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s : f : \Z \to \N, defined by: : f(n) = m if, and only if, m^2 = n, m \in \N, n \in \Z, then f(n) is only defined if n is a perfect square (that is, 0, 1, 4, 9, 16, \ldots). So f(25) = 5 but f(26) is undefined.


Basic concepts

A partial function arises from the consideration of maps between two sets and that may not be defined on the entire set . A common example is the square root operation on the real numbers \mathbb: because negative real numbers do not have real square roots, the operation can be viewed as a partial function from \mathbb to \mathbb. The ''domain of definition'' of a partial function is the subset of on which the partial function is defined; in this case, the partial function may also be viewed as a function from to . In the example of the square root operation, the set consists of the nonnegative real numbers [0, +\infty). The notion of partial function is particularly convenient when the exact domain of definition is unknown or even unknowable. For a computer-science example of the latter, see ''Halting problem''. In case the domain of definition is equal to the whole set , the partial function is said to be ''total''. Thus, total partial functions from to coincide with functions from to . Many properties of functions can be extended in an appropriate sense of partial functions. A partial function is said to be
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
,
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
, or
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
when the function given by the restriction of the partial function to its domain of definition is injective, surjective, bijective respectively. Because a function is trivially surjective when restricted to its image, the term
partial bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
denotes a partial function which is injective. 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. The notion of
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Tran ...
can be generalized to partial functions as well. A partial transformation is a function f : A \rightharpoonup B, where both A and B are
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of some set X.


Function spaces

For convenience, denote the set of all partial functions f : X \rightharpoonup Y from a set X to a set Y by \rightharpoonup Y This set is the union of the sets of functions defined on subsets of X with same codomain Y: : \rightharpoonup Y= \bigcup_ \to Y the latter also written as \bigcup_ Y^D. In finite case, its cardinality is : , \rightharpoonup Y = (, Y, + 1)^, because any partial function can be extended to a function by any fixed value c not contained in Y, so that the codomain is Y \cup \, 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 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.


Natural logarithm

Consider the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
function mapping the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s 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 In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
(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

Subtraction of
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
(non-negative
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
) can be viewed as a partial function: : f : \N \times \N \rightharpoonup \N : f(x,y) = x - y. It is defined only when x \geq y.


Bottom element

In denotational semantics a partial function is considered as returning the
bottom element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an eleme ...
when it is undefined. In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
a partial function corresponds to a subroutine that raises an exception or loops forever. The
IEEE floating point The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in ...
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 A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
where function parameters are statically typed, a function may be defined as a partial function because the language's
type system In computer programming, a type system is a logical system comprising a set of rules that assigns a property called a type to every "term" (a word, phrase, or other set of symbols). Usually the terms are various constructs of a computer progra ...
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 In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of t ...
, the composition operation \circ \;:\; \hom(C) \times \hom(C) \to \hom(C) is a function if and only if \operatorname(C) has one element. The reason for this is that two morphisms f : X \to Y and g : U \to V can only be composed as g \circ f if Y = U, that is, the codomain of f must equal the domain of g. The category of sets and partial functions is
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...
to but not
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
with the category of
pointed set In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point, also spelled basepoint. Maps between pointed sets (X, x_0) and (Y, y_0) – called based ...
s 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 In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Al ...
) and in
theoretical computer science Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumsc ...
." The category of sets and partial bijections is equivalent to its dual. It is the prototypical inverse category.


In abstract algebra

Partial algebra In abstract algebra, a partial algebra is a generalization of universal algebra to partial operations. Example(s) * partial groupoid * field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultu ...
generalizes the notion of
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of stu ...
to partial operations. An example would be a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, in which the multiplicative inversion is the only proper partial operation (because
division by zero In mathematics, division by zero is division (mathematics), division where the divisor (denominator) is 0, zero. Such a division can be formally expression (mathematics), expressed as \tfrac, where is the dividend (numerator). In ordinary ari ...
is not defined). The set of all partial functions (partial
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Tran ...
s) on a given base set, X, forms a
regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
called the semigroup of all partial transformations (or the partial transformation semigroup on X), typically denoted by \mathcal_X. The set of all partial bijections on X forms the
symmetric inverse semigroup __NOTOC__ In abstract algebra, the set of all partial bijections on a set ''X'' ( one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup (actually a monoid) on ''X''. The conventional notation for the ...
.


Charts and atlases for manifolds and fiber bundles

Charts in the atlases which specify the structure of
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s and
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s 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 In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
, 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.


See also

* * *


References

* Martin Davis (1958), ''Computability and Unsolvability'', McGraw–Hill Book Company, Inc, New York. Republished by Dover in 1982. . *
Stephen Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
(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. Mathematical relations Functions and mappings