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functions: specification
and definition
This section describes precisely what is meant by “function”.
¨ We start
by giving a specification of “function”.
¨ The
sections on notation
and usage
spell out many of the ways a definition of a particular function can be given.
¨ After
that, we get into the technicalities of the mathematical definitions
of the general concept of
function. It is necessary to have these
definitions for some purposes, but usually the specifications are enough. Things get complicated because there are two
different definitions of “function” in common use, even though this difference
generally causes little trouble!
Specification of function
We start by giving a specification of function, then later get into the technicalities of the definitions.
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Examples
I will use two running examples throughout this discussion:
¨ F is the function defined on the set as follows:
. This is the
function called “Finite” in the chapter on examples
of functions.
¨ G is the real-valued
function defined by the formula .
Specification: functions
A function (
is the letter phi) is a mathematical
object which determines
and is completely determined by the following data:
¨
has a domain, which
is a set. The domain may be denoted by dom
.
¨
has a codomain,
which is also a set and may be denoted by cod
.
¨
For each element a of the domain of ,
has a value at a.
a) The value of at
a is completely determined by a and F .
b) The value of at
a must be an element of the codomain
of F.
c) The value of at
a is denoted by
(a).
d) a is called the argument or independent variable or input to F.
e) (a) is the value or dependent
variable or output. (See dependency relation.)
¨
The operation of finding (a)
given
and a
is called evaluation or application.
Examples
¨ The definition of the finite function F specifies that the domain is the
set . Its codomain is not specified, but must
include the set {1,2,3}. The value of F at 3, for example, is 2, because the definition says that F(2) = 3.
¨
The definition of G above
does not specify the domain or the codomain.
The convention in the case of functions on the real numbers is to take the domain to be all real numbers at which the formula is defined. In this case, that is every real number, so the domain is . The codomain must include all real numbers greater than or equal to 4.
(Why?)
¨ The Split function is explicitly given the domain the closed interval [0,1] . Its codomain is not given. It must include [0,1].
¨
The Sine Blur function is explicitly given
the domain of positive real numbers. Its
codomain must include the closed interval [1,
1] since the sine function takes every value in that interval.
Notation
Arrow Notation
Examples
¨
For we could write
(choosing the codomain to be all of
).
¨
For the finite
function we could write .
Usage
The expression can be used as name or as an independent
sentence, so it has
context-sensitive pronunciation.
¨
Standing
alone, the expression may be read aloud this way: “ is a function from A to B”
or “
goes from A
to B”.
¨
The expression may occur embedded
in a sentence, as in “Let be a differentiable function.” This may be read “Let
from A to B be a
differentiable function.”
Warnings
¨
The statement by itself does
not determine the function
. It says only that
its name is
,
its domain is the set A, and its
codomain is the set B. For example,
for
,
and a gazillion other functions we may have
.
¨
You should distinguish between , which is the name of the function and
(a),
which is the value of
at an input value a. Nevertheless, such a
function is very commonly referred to as
(x). For functions given by formulas, this notation
has the value of telling you what letter will be used for the input variable.
Ways of defining a function
Table
A function defined on a finite set may be given by a table; for example, the function F defined above.
Formula
A function may be defined by giving an algebraic expression
(its formula) that determines its
value. An example is the function G above. The expression may be called the defining equation of the
function, and
its defining expression. The defining expression may (controversially)
be used as its name.
Algorithm
Geometric definition
The circumference function C(r) could be defined this
way: C(r) is the circumference of a circle with
radius r.
Of course it can be given by a formula, too: . Be clear that the geometric definition is just as precise and exact a
definition as the formula is.
Barred arrow notation
¨
The function G defined above could be written .
¨ The barred arrow goes from the input to a formula for the output.
¨ The straight arrow goes from domain to codomain.
¨
It is less wordy than “G is the function defined defined on the reals by the formula .”
¨
It is useful when there are parameters,
as in for example .
¨
It allows anonymous structural
notation in the way that matrix notation does: . I wish more people would use this when they
want to refer to a particular function just once. It gets rid of the name G, thus lowering the burden on the reader.
With finite functions
A variant of barred arrow notation is to define functions on
finite sets element by element. For
example the finite
function F could be
defined by: .
Terminology
Map, mapping
Some texts use map or mapping to mean "function”. Others distinguish between maps and functions, for example requiring that a mapping be a continuous function, or requiring that a map have a specified codomain (in other words use the stricter definition of function.) Other uses of the word are described in Wikipedia.
Functional
The word functional is used as a noun to denote some special class of functions. The most common use seems to be to denote a function whose domain consists of elements of some function space over a field and whose values are elements of the field. (This claim is not based on lexicographical research and needs to be investigated further.) But the word is used in other senses, as well. See the Wikipedia entry.
Transformation
Functions may be called transformations. The word is commonly used when the domain and the codomain are the same, but that is not the case when it is used in the phrase linear transformation. (This claim is not based on lexicographical research.)
Multivalued
function
Partial function
Mathematical definitions of function
This section gives the definition(s) of function usually given in textbooks.
In the nineteenth century, mathematicians realized that it
was necessary for some purposes (particularly harmonic analysis) to
give a mathematical definition of the concept of
function. A stricter version of this
definition turned out to be necessary in algebraic topology and other fields,
so now there are two
nonequivalent definitions in common use.
The difference
between these definitions causes much less trouble than you would think!
To state these
two definitions we need a preliminary idea.
The functional property
Definition
A set R of ordered pairs has the functional property if two pairs in R with the same first coordinate have to have the same second coordinate (which means they are the same pair).
Examples
¨ The set {(1,2), (2,4), (3,2), (5,8)} has the functional property, since no two different pairs have the same first coordinate.
¨ The set {(1,2), (2,4), (3,2), (2,8)} does not have the functional property. There are two different pairs with first
coordinate 2.
Note that in
both sets there are two different pairs with the same second coordinate. This is irrelevant to the
functional property.
How to think about the functional property
The point of the functional property is
that for any pair in the set of ordered pairs, the first coordinate determines what
the second one is. That’s why you can write “G(x)” for any x in the domain
of G and not be ambiguous.
Less strict definition of function
This definition (and the stricter
one as well) is an
abstraction of
the naïve concept of function. It is based on one particular aspect (the graph) of all the images and metaphors involved
in our picture of a function.
Definition
¨ A function F is a set of ordered pairs with the functional property. The set of ordered pairs is also called the graph of F.
More about the graph.
¨ If F is a function, the domain of F is the set of first coordinates of all the pairs in F.
¨ If then F(x) is the second coordinate of the only ordered pair in F whose first
coordinate is x.
Example
The set {(1,2), (2,4), (3,2), (5,8)} has the functional
property, so it is a function. Call it F. Then
its domain is {1,2,3,5} and F(1) = 2
and F(2) = 4. F(4)
is not defined because there is no ordered pair in F beginning with 4 (hence 4 is not in dom F.)
Example
The function ordinarily described as “the function ” fits the definition of function given
here. You define F to be the set
,
which is normally called the graph of F. This set has the functional property
because if x is any real number, the
formula
defines a specific real number. (If you plug in a real number, you get one real number, not several.) Example:
¨
if x = 0, then ,
so the pair (0, 5) is “in F.”
¨
if x = 1, then .
¨ if x = 2,
then
.
No other pair whose first coordinate
is 2
is in F, only (
-2,
5). That is because when you plug
2
into the formula
,
you get 5 and nothing else. Of course,
(0, 5) is in F, but that does not
contradict the functional property. (0,
5) and (
2,
5) have the same second
coordinate, which is OK.
Stricter definition of function
A function F is a mathematical structure consisting of the following objects:
¨ A set called the domain of F, denoted by dom F.
¨
A set called the codomain of F,
denoted by cod F.
¨ A set of ordered pairs called the graph of F, with the following properties:
f) dom F is the set of all first coordinates of pairs in the graph of F.
g) Every second coordinate of a pair in the graph of F is in cod F (but cod F may contain other elements).
h) The graph of F has the functional property.
The notation means:
¨ F is a function,
¨
A = dom F,
and
¨ B = cod F.
Note that this notation does not tell you which function F is, only that it has domain A and codomain B.
Examples
¨
Let F have graph {(1,2), (2,4), (3,2),
(5,8)} (previously discussed here),
and define A = {1, 2, 3, 5} and B = {2, 4, 8}. Then is a function.
¨
Let F have graph {(1,2), (2,4), (3,2),
(5,8)} (same as above), and define A = {1, 2, 3, 5} and C = {2, 4, 8, 9, 11, ,
3/2}. Then
is a (slightly ridiculous) function. Note that all the second coordinates of the
graph are in C, along with a bunch of
miscellaneous suspicious characters that are not second coordinates
of pairs in the graph. See equality of functions.
¨
With the same definition of the
graph and of A, is a function.
¨
Using the same definition of the
graph, let D = {1, 2, 5} and E = {1, 2, 3, 4, 5}. Then
neither nor
is a function, because neither D nor E has exactly the same elements as the first
coordinates of the graph of F.
Identity and inclusion
Suppose we have two sets A and B with .
¨
The identity function on A is the function defined by
for all
. (Many authors call it
).
¨
The inclusion function from
A to B is the function defined by
for all
. Note that there is a different function for
each pair of sets A and B for which
. Some authors call it
or
.
Remark
The identity
function and an inclusion function for the same set A have exactly the same graph, namely . Thus according to the less strict definition, they are the same
function.
How to think about the definitions of function
Finish this
Equality of functions To
be written
Appendices
“Formulas determine the
output”
“The formula defines a specific real number. (If you plug in a real number, you get one
real number, not several.)
This statement is true of many algebraic formulas, but not all.
Example
The formula determines a specific real number for any real
number x except x =
1. Youo can still use it to define a function, but the
domain has to exclude
1 (at least).
For example, you can talk about the function
defined by
, where
denotes the set of
nonnegative real numbers.
Example
The
formula determines two specific real numbers for each real number x (with one
exception!). It is not suitable for
definining a function. Back
Functions
in math, logic and computing science
Until late in the nineteenth century, functions were usually
thought of as defined by formulas. Problems arose in the theory of harmonic analysis
which made mathematicians require a more general notion of function. The definitions of function given here is the
modern version of that more general concept. It replaces the algorithmic and dynamic idea of
a function as a way of computing an output value given an input value by the static, abstract
concept of a function as having a domain, a codomain and a value lying in the codomain for each element of the domain. Of
course, often a definition by formula will give a function in this modern sense. However, there is no requirement
that a function be given by a formula.
The modern concept of function has been obtained from the formula-based idea by abstracting basic properties the old concept had (specifically properties of the graph) and using them as the basis of the new definition. The concept of function as a formula never disappeared entirely, but was studied mostly by logicians who generalized it to the study of function-as-algorithm. (This is an oversimplification of history.) Of course, the study of algorithms is one of the central topics of modern computing science, so the notion of function-as-formula (more generally, function-as-algorithm) has achieved a new importance in recent years.
Nevertheless, computer science needs the abstract definition of function given here. Functions such as sine may be (and quite often are) programmed to look up their values in a table instead of calculating them by a formula, an arrangement which gains speed at the expense of using more memory. The example of a finite function given above is a baby example of a table look-up function.
Appendix
Question: Why must the codomain of G include every real number greater than or equal to 4?
Answer: Because the minimum value of G is 4 (that occurs at 1)
and it takes on every value bigger than 4. Back