# Function and codomain

I recently posted the following information in the talk page of the Wikipedia article on functions, where they were arguing about whether "function" means a set of ordered pairs with the functional property or a structure with a domain $D$, a codomain $C$, and a graph $G$ which is a subset of $D\times C$ with the functional property.

I collected data from some math books published since 2000 that contain a definition of function; they are listed below.  In this list, "typed" means  function was defined as going from a set A to a set B, A was called the domain, and B was not given a name. If "typed" is followed by a word (codomain, range or target) that was the name given the codomain. One book defined a function essentially as a partial function. Some that did not name the codomain defined "range" in the sense of image. Some of them emphasized that the range/image need not be the same as the codomain.

As far as I know, none of these books said that if two functions had the same domain and the same graph but different codomains they had to be different functions.  But I didn't read any of them extensively.

My impression is that modern mathematical writing at least at college level does distinguish the domain, codomain, and image/range of a function, not always providing a word to refer to the codomain.

If the page number as a question mark after it that means I got the biblio data for the book from Amazon and the page number from Google books, which doesn't give the edition number, so it might be different.

I did not look for books by logicians or computing scientists.  My experience is that logicians tend to use partial functions and modern computing scientists generally require the codomain to be specified.

Opinion:  If you don't distinguish functions as different if they have different codomains, you lose some basic intuition (a function is a map) and you mess up common terminology.  For example the only function from {1} to {1} is the identity function, and is surjective.  The function from {1} to the set of real numbers (which is a point on the real line) is not the identity function and is not surjective.

THE LIST

Mathematics for Secondary School Teachers
By Elizabeth G. Bremigan, Ralph J. Bremigan, John D. Lorch, MAA 2011
p. 6 (typed)

Oxford Concise Dictionary of Mathematics, ed. Christopher Clapham and James Nicholson,  Oxford University Press, 4th ed., 2009.
p. 184, (typed, codomain)

Math and Math-in-school: Changes in the Treatment of the Function Concept in …
By Kyle M. Cochran, Proquest, 2011
p74  (partial function)

Discrete Mathematics: An Introduction to Mathematical Reasoning
By Susanna S. Epp, 4th edition, Cengage Learning, 2010
p. 294? (typed, co-domain)

Teaching Mathematics in Grades 6 – 12: Developing Research-Based …
By Randall E. Groth, SAGE, 2011
p236 (typed, codomain)

Essentials of Mathematics, by Margie Hale, MAA, 2003.
p. 38 (typed, target).

Elements of Advanced Mathematics
By Steven G. Krantz, 3rd ed., Chapman and Hall, 2012
p79? (typed, range)

Bridge to Abstract Mathematics
By Ralph W. Oberste-Vorth, Aristides Mouzakitis, Bonita A. Lawrence, MAA 2012
p76 (typed, codomain)

The Road to Reality by Roger Penrose, Knopf, 2005.
p. 104 (typed, target)

Precalculus: Mathematics for Calculus
By James Stewart, Lothar Redlin, Saleem Watson, Cengage, 2011
p. 143.  (typed)

The Mathematics that Every Secondary School Math Teacher Needs to Know
By Alan Sultan, Alice F. Artzt , Routledge, 2010.
p.400 (typed)

# More about the definition of function

Maya Incaand commented on my post Definition of "function":

Why did you decide against "two inequivalent descriptions in common use"?  Is it no longer true?

This question concerns [1], which is a draft article.  I have not promoted it to the standard article in abstractmath because I am not satisfied with some things in it.

More specifically, there really are two inequivalent descriptions in common use.  This is stated by the article, buried in the text, but if you read the beginning, you get the impression that there is only one specification.  I waffled, in other words, and I expect to rewrite the beginning to make things clearer.

Below are the two main definitions you see in university courses taken by math majors and grad students.  A functional relation has the property that no two distinct ordered pairs have the same first element.

Strict definition: A function consists of a functional relation with specified codomain (the domain is then defined to be the set of first elements of pairs in the relation).  Thus if $A$ and $B$ are sets and $A\subseteq B$, then the identity function $1_A:A\to A$ and the inclusion function $i:A\to B$  are two different functions.

Relational definition: A function is a functional relation.  Then the identity and inclusion functions are the same function.  This means that a function and its graph are the same thing (discussed in the draft article).

These definitions are subject to variations:

Variations in the strict definition: Some authors use "range" for "codomain" in the definition, and some don't make it clear that two functions with the same functional relation but different codomains are different functions.

Variations in the relational definition: Most such definitions state explicitly that the domain and range are determined by the relation (the set of first coordinates and the set of second coordinates).

### Formalism

There are many other variations in the formalism used in the definition.  For example, the strict definition can be formalized (as in Wikipedia) as an ordered triple $(A, B, f)$ where $A$ and $B$ are sets and $f$ is a functional relation with the property thar every element of $A$ is the first element of an ordered pair in the relation.

You could of course talk about an ordered triple $(A,f,B)$ blah blah.  Such definitions introduce arbitrary constructions that have properties irrelevant to the concept of function.  Would you ever say that the second element of the function $f(x)=x+1$ on the reals is the set of real numbers?  (Of course, if you used the formalism $(A,f,B)$ you would have to say the second element of the function is its graph! )

It is that kind of thing that led me to use a specification instead of a definition.  If you pay attention to such irrelevant formalism there seems to be many definitions of function.  In fact, at the university level there are only two, the strict definition and the relational definition.  The usage varies by discipline and age.  Younger mathematicians are more likely to use the strict definition.  Topologists use the strict definition more often than analysts (I think).

### Usage

There is also variation in usage.

• Most authors don't tell you which definition they use, and it often doesn't matter anyway.
• If an author defines a function using a formula, there is commonly an implicit assumption that the domain includes everything for which the formula is well-defined.  (The "everything" may be modified by referring to it as an integer, real, or complex function.)

### Definitions of function on the web

Below are some definitions of function that appear on the web.  I have excluded most definitions aimed at calculus students or below; they often assume you are talking about numbers and formulas.  I have not surveyed textbooks and research papers.  That would have to be done for a proper scholarly article about mathematical usage of "function". But most younger people get their knowledge from the web anyway.

1. Abstractmath draft article: Functions: Specification and Definition.  (Note:  Right now you can't get to this from the Table of Contents; you have to click the preceding link.)
2. Gyre&Gimble post: Definition of "function"
3. Intmath discussion of function  Function as functional relation between numbers, with induced domain and range.
4. Mathworld definition of function Functional-relation definition.  Defines $F:A\to B$ in a way that requires $B$ to be the image.
5. Planet Math definition of function Strict definition.
6. Prime Encyclopedia of Mathematics Functional-relation definition.
7. Springer Encyclopedia of Math definition of function  Strict definition, except not clear if different codomains mean different functions.
8. Wikipedia definition of function Discusses both definitions.
9. Wisconsin Department of Public Instruction Definition of function  Function as functional relation.

# The meaning of the word “superposition”

This is from the Wikipedia article on Hilbert's 13th Problem as it was on 31 March 2012:

[Hilbert’s 13th Problem suggests this] question: can every continuous function of three variables be expressed as a composition  of finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by Vladimir Arnold, then only nineteen years old and a student of Andrey Kolmogorov. Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering Hilbert's question.

In their paper A relation between multidimensional data compression and Hilbert’s 13th  problem,  Masahiro Yamada and Shigeo Akashi describe an example of Arnold's theorem this way:

Let $f ( \cdot , \cdot, \cdot )$ be the function of three variable defined as $f(x, y, z)=xy+yz+zx$, $x ,y , z\in \mathbb{C}$ . Then, we can easily prove that there do not exist functions of two variables $g(\cdot , \cdot )$ , $u(\cdot, \cdot)$ and $v(\cdot , \cdot )$ satisfying the following equality: $f(x, y, z)=g(u(x, y),v(x, z)) , x , y , z\in \mathbb{C}$ . This result shows us that $f$ cannot be represented any 1-time nested superposition constructed from three complex-valued functions of two variables. But it is clear that the following equality holds: $f(x, y, z)=x(y+z)+(yz)$ , $x,y,z\in \mathbb{C}$ . This result shows us that $f$ can be represented as a 2-time nested superposition.

The article about superposition in All about circuits says:

The strategy used in the Superposition Theorem is to eliminate all but one source of power within a network at a time, using series/parallel analysis to determine voltage drops (and/or currents) within the modified network for each power source separately. Then, once voltage drops and/or currents have been determined for each power source working separately, the values are all “superimposed” on top of each other (added algebraically) to find the actual voltage drops/currents with all sources active.

Superposition Theorem in Wikipedia:

The superposition theorem for electrical circuits states that for a linear system the response (Voltage or Current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are replaced by their internal impedances.

Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system — such as an electron — exists partly in all its particular, theoretically possible states (or, configuration of its properties) simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations (as described in interpretation of quantum mechanics).

Mathematically, it refers to a property of solutions to the Schrödinger equation; since theSchrödinger equation is linear, any linear combination of solutions to a particular equation will also be a solution of it. Such solutions are often made to be orthogonal (i.e. the vectors are at right-angles to each other), such as the energy levels of an electron. By doing so the overlap energy of the states is nullified, and the expectation value of an operator (any superposition state) is the expectation value of the operator in the individual states, multiplied by the fraction of the superposition state that is "in" that state

The CIO midmarket site says much the same thing as the first paragraph of the Wikipedia Quantum Superposition entry but does not mention the stuff in the second paragraph.

In particular, the  Yamada & Akashi article describes the way the functions of two variables are put together as "superposition", whereas the Wikipedia article on Hilbert's 13th calls it composition.  Of course, superposition in the sense of the Superposition Principle is a composition of multivalued functions with the top function being addition.  Both of Yamada & Akashi's examples have addition at the top.  But the Arnold theorem allows any continuous function at the top (and anywhere else in the composite).

So one question is: is the word "superposition" ever used for general composition of multivariable functions? This requires the kind of research I proposed in the introduction of The Handbook of Mathematical Discourse, which I am not about to do myself.

The first Wikipedia article above uses "composition" where I would use "composite".  This is part of a general phenomenon of using the operation name for the result of the operation; for examples, students, even college students, sometimes refer to the "plus of 2 and 3" instead of the "sum of 2 and 3". (See "name and value" in abstractmath.org.)  Using "composite" for "composition" is analogous to this, although the analogy is not perfect.  This may be a change in progress in the language which simplifies things without doing much harm.  Even so, I am irritated when "composition" is used for "composite".

Quantum superposition seems to be a separate idea.  The second paragraph of the Wikipedia article on quantum superposition probably explains the use of the word in quantum mechanics.