Partial function

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In mathematics, a partial function is a binary relation over two sets that associates to every element of the first set at most one element of the second set. It generalizes the concept of a function by not requiring every element of the first set to be associated to at least one element of the second set. Consequently, the domain of definition of a partial function can be a proper subset of its domain, contrary to a function for which the two sets always coincide.

A partial function f over X and Y is sometimes written as f: XY, f: XY, or f: XY[citation needed]. Partial functions are often used when their exact domain of definition is not known (for example in computability theory, general recursive functions are partial functions from the integers to the integers, and there cannot be any algorithm for deciding whether such partial functions are functions). In real and complex analysis, a partial function is generally called simply a function.

Specifically, we will say that for any xX, either:

  • f(x) = yY (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

f: ZZ
f(n) = √n.

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

Basic conceptsEdit

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

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.

Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective.[1]

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 can be generalized to partial functions as well. A partial transformation is a function f: AB, where both A and B are subsets of some set X.[1]

FunctionEdit

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.

Function spacesEdit

The set of all partial functions f: XY from a set X to a set Y, denoted by [XY], is the union of all functions defined on subsets of X with same codomain Y:

 

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 c not contained in Y, so that the codomain is Y ∪ {c}, an operation which is injective (unique and invertible by restriction).

Discussion and examplesEdit

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.

Natural logarithmEdit

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 numbersEdit

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

 
 

It is defined only when  .

Bottom elementEdit

In denotational semantics a partial function is considered as returning the bottom element when it is undefined.

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 theoryEdit

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.[2] 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."[3]

The category of sets and partial bijections is equivalent to its dual.[4] It is the prototypical inverse category.[5]

In abstract algebraEdit

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).[6]

The set of all partial functions (partial transformations) on a given base set, X, forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on X), typically denoted by  .[7][8][9] The set of all partial bijections on X forms the symmetric inverse semigroup.[7][8]

Charts and atlases for manifolds and fiber bundlesEdit

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.

See alsoEdit

ReferencesEdit

  1. ^ a b Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.
  2. ^ Lutz Schröder (2001). "Categories: a free tour". In Jürgen Koslowski and Austin Melton (ed.). Categorical Perspectives. Springer Science & Business Media. p. 10. ISBN 978-0-8176-4186-3.
  3. ^ Neal Koblitz; B. Zilber; Yu. I. Manin (2009). A Course in Mathematical Logic for Mathematicians. Springer Science & Business Media. p. 290. ISBN 978-1-4419-0615-1.
  4. ^ Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge University Press. p. 289. ISBN 978-0-521-44179-7.
  5. ^ Marco Grandis (2012). Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups. World Scientific. p. 55. ISBN 978-981-4407-06-9.
  6. ^ Peter Burmeister (1993). "Partial algebras – an introductory survey". In Ivo G. Rosenberg and Gert Sabidussi (eds.). Algebras and Orders. Springer Science & Business Media. ISBN 978-0-7923-2143-9.CS1 maint: uses editors parameter (link)
  7. ^ a b Alfred Hoblitzelle Clifford; G. B. Preston (1967). The Algebraic Theory of Semigroups. Volume II. American Mathematical Soc. p. xii. ISBN 978-0-8218-0272-4.
  8. ^ a b Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press, Incorporated. p. 4. ISBN 978-0-19-853577-5.
  9. ^ Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. pp. 16 and 24. ISBN 978-1-84800-281-4.
  • 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.