Function

A function is a formal operation that maps every member of a function domain to a member of a function range.

• AKA: ƒ, F.
• Context:
  • It must have a Function Input (composed of one or more Function Parameters) from the Function Domain).
  • It must have a Function Output (which is a member of the Function Range).
  • It can be Formally Defined as:
    • let D and R be Sets. A function ƒ is a Binary Relation defined for each Element in B and some Element of D.
  • It can be: a Unary Function, a Binary Function, or an n-Ary Function.
  • It can be, depending on what proportion of the Domain Range is mapped:
    • a Total Function, or a Partial Function.
  • It can be, depending on the Type of the Function Range.
    • a Value-Output Function
      • a Discrete Value-Output Function, such as a Relation, Classification Function, or an Ordinal-Output Function.
      • a Continuous-Output Function, such as Number-Output Function.
    • a Set-Output Function, or a Multiset-Output Function,
    • a Sequence-Output Function, or a String-Output Function,
    • a Tuple-Output Function, or a Vector-Output Function.
  • It can be, depending on the Type of the Function Domain.
    • a Class-Input Function, such as a Binary-Input Function.
    • an Ordinal-Input Fuction or a Number-Input Function,
    • a Set-Input Function, or a Multiset-Input Function,
    • a Sequence-Input Function, or a String-Input Function,
    • a Tuple-Input Function, or a Vector-Input Function.
  • It can be:
    • a Bijective Function.
    • an Injective Function.
    • a Surjective Function.
  • It can be:
    • an Associative Function.
    • a Commutative Function.
    • a Distributive Function.
  • It can be:
    • a Scalar Function.
    • a Set Function, Multiset Function.
    • a String Function, Sequence Function.
    • a Tuple Function, Vector Function.
  • It can require that the Function Domain and [Function Range]] be Distinct Sets.
  • It can be a Predictive Function with a Probability Value.
• Example(s):
  • an Algegraic Function, composed of Algebraic Expressions.
  • A Metric Function.
  • A Cosine Function.
  • A Vector Length Function.
  • A Vector Dot Product Function.
• Counter-Example(s):
  • a Real Number.
  • a Symbol.
  • an Informal Operation.
• See: Algorithm, Function Parameter, Ordered Pair, Malfunction.

References

2009

• (Wikipedia, 2009) http://en.wikipedia.org/wiki/Function_(mathematics)
  • In mathematics, a function is a relation between a given set of elements called the domain and a set of elements called the codomain. The function associates each element in the domain with exactly one element in the codomain. The elements so related can be any kind of thing (words, objects, qualities) but are typically mathematical quantities, such as real numbers.
  • An example of a function with domain {A,B,C} and codomain {1,2,3} associates A with 1, B with 2, and C with 3. An example of a function with the real numbers as both its domain and codomain is the function f(x) = 2x, which associates every real number with the real number twice as big. In this case, we can write f(5) = 10.
• (Wikipedia, 2009) http://en.wikipedia.org/wiki/Function_(Programming)
  • In computer science, a subroutine (function, method, procedure, or subprogram) is a portion of code within a larger program, which performs a specific task and can be relatively independent of the remaining code. ...
• http://planetmath.org/encyclopedia/Range2.html
  • A function is a triplet (f,A,B) where:
    • 1. A is a set (called the domain of the function).
    • 2. B is a set (called the codomain of the function).
    • 3. f is a binary relation between A and B.
    • 4. For every a in A, there exists b in B such that (a,b) in f.
    • 5. If a in A, b_1,b_2 in B, and (a,b_1) in f and (a,b_2) in f, then b_1 = b_2.
  • The triplet (f,A,B) is usually written with the specialized notation f: A->B. This notation visually conveys the fact that f maps elements of A into elements of B.
• CYC Glossary http://www.cyc.com/cycdoc/ref/glossary.html
  • function: A function (in the mathematical sense) is a relation such that for each thing in its domain (the universe of things it can be applied to), there is a single thing in its range (the universe of results it can have) such that the relation holds between them. In CycL, functions are denoted by certain constants. These constants are referred to as "function-denoting constants, "CycL functions," or sometimes just "functions." CycL functions can be applied to arguments to form non-atomic terms, which can serve as arguments to a predicate just as other terms can. There are more details about CycL functions in the Functions section of "The Syntax of Cycl".
• http://www.isi.edu/~hobbs/bgt-sequences.text
  • A function f from a set s1 to a set s2 is a set of pairs where each element of s1 occurs exactly once as the first element in a pair, and every element of s2 occurs as a second element of at least one pair. We will call this predicate "function0" because in Chapter B7 we will extend it to functions from scales to scales as well.
    • (forall (f s1 s2) (5) (iff (function0 f s1 s2) (and (set s1)(set s2) (forall (p) (if (member p f) (and (pair p) (forall (x y) (if (and (first x p)(second y p)) (and (member x s1) (member y s2))))))) (forall (x) (if (member x s1) (exists (p) (and (member p f)(first x p))))) (forall (p1 p2 x) (if (and (first x p1)(first x p2) (member p1 f)(member p2 f)) (equal p1 p2))) (forall (y) (if (member y s2) (exists (p) (and (member p f)(second y p))))))))
  • Lines 4-10 of this definition say that a function is a set of pairs, the first elements of which come from s1 and the second elements of which come from s2. Lines 11-14 say that there is a value of the function for every element of s1. Lines 15-18 say that that value is unique. Lines 19-22 say that every element of s2 is a value of some member of s1 under the function f.
• http://ltcconline.net/greenl/courses/202/vectorIntegration/vectorFields.htm#fields
  • We have now seen many types of functions. They are characterized by the domain and the range.
  • Below is a list of some of the functions that we have encountered so far.
    

    Domain	Range	Name
    R	R	One variable Function
    R	R2	Parametric Equations
    R2	R	Function of 2 Variables
    R	Vectors	Vector Valued Function

• http://www.math.com/tables/algebra/functions/index.htm
  • Definition: A function is a relation from a domain set to a range set, where each element of the domain set is related to exactly one element of the range set.
  • An equivalent definition: A function (f) is a relation from a set A to a set B (denoted f: A�B), such that for each element in the domain of A (Dom(A)), the f-relative set of A (f(A)) contains exactly one element.