Tag Archives: function

Definition of function

Note: This is a revision of the article on specification and definition of functions from abstractmath.org. Many of the links in this article take you to other articles in abstractmath.org.

A function is a mathematical object.

To deal with functions as a math object, you need a precise definition of “function”. That is what this article gives you.

  • The article starts by giving a specification of “function”.
  • After that, we get into the technicalities of the
    definitions of the general concept of function.
  • Things get complicated because there are several inequivalent definitions of “function” in common use.

Specification of “function”

A function $f$ is a mathematical object which determines and is completely determined by the following data:


  • (DOM) $f$ has a domain, which is a set. The domain may be denoted by $\text{dom} f$.
  • (COD) $f$ has a codomain, which is also a set and may be denoted by $\text{cod} f$.
  • (VAL) For each element $a$ of the domain of $f$, $f$ has a value at $a$.
  • (FP) The value of $f$ at $a$ is
    completely determined by $a$ and $f$.
  • (VIC) The value of $f$ at $a$ must be an element of the codomain of $f$.

  • The value of $f$ at $a$ is most cohttp://www.abstractmath.org/MM/MMonly written $f(a)$, but see Functions: Notation and Terminology.
  • To evaluate $f$ at $a$ means to determine $f(a)$. The two examples of functions below show that different functions may have different strategies for evaluating them.
  • In the expression “$f(a)$”, $a$ is called the input or (old-fashioned) argument of $f$.
  • “FP” means functional property.
  • “VIC” means “value in codomain”.

Examples

I give two examples here. The examples of functions chapter contains many other examples.

A finite function

Let $F$ be the function defined on the set $\left\{\text{a},\text{b},\text{c},\text{d}\right\}$ as follows: \[F(\text{a})=\text{a},\,\,\,F(\text{b})=\text{c},\,\,\,F(\text{c})=\text{c},\,\,\,F(\text{d})=\text{b}\]In this definition, $\text{a},\text{b},\text{c},\text{d}$ are letters of the alphabet, not variables. This is the function called “Finite” in the chapter on examples of functions.

  • The definition of $F$ says “$F$ is defined on the set $\left\{\text{a},\,\text{b},\,\text{c},\,\text{d} \right\}$”. The phrase “is defined on”
    means that the domain is that set. That is standard terminology.
  • The value of $F$ at each element of the domain is given explicitly. The value at
    $\text{b}$, for example, is $\text{c}$, because the definition says that $F(\text{b}) = \text{c}$. No other reason needs to be given. Mathematical definitions can be arbitrary.
  • The codomain of $F$ is not specified, but must include the set $\{\text{a},\text{b},\text{c}\}$. The codomain of a function is often not specified when it is not important, which is most of the time in freshman calculus (for example).
  • The diagram below shows how $F$ obeys the rule that the value of an element $x$ in the domain is completely determined by $x$ and $F$.
  • If two arrows had started from the same element of the domain, then $F$ would not be a function. (It would be a multivalued function).
  • If there were an element of the domain that no arrow started from, it $F$ would not be a function. (It would be a partial function.)
  • In this example, to evaluate $F$ at $b$ (to determine the value of $F$ at $b$) means to look at the definition of $F$, which says among other things that the value is $c$ (or alternatively, look at the diagram above and see what letter the arrow starting at $b$ points to). In this case, “evaluation” does not imply calculating a formula.

A real-valued function

Let $G$ be the real-valued function defined by the formula $G(x)={{x}^{2}}+2x+5$.

  • The definition of $G$ gives the value at each element of the domain by a formula. The value at $3$, for example, is obtained by calculating \[G(3)=3^2+2\cdot3+5=20\]
  • The definition of $G$
    does not specify the domain. The convention in the case of functions defined on the real numbers by a formula 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 $\mathbb{R}$.
  • The definition of $G$ does not specify the codomain, either. However, the codomain must include all real numbers greater than or equal to $4$. (Why?)
  • So if an author wrote, “Let $H(x)=\frac{1}{x}$”, the domain would be the set of all real numbers except $0$. But a careful author would write, “Let $H(x)=\frac{1}{x}$ ($x\neq0$).”

What the specification means

  • The specification guarantees that a function satisfies all five of the properties listed.
  • The specification does not define a mathematical structure in the way mathematical structures have been defined in the past: In particular, it does not require a function to be one or more sets with structure.
  • Even so, it is useful to have the specification, because:

    Many mathematical definitions
    introduce extraneous technical elements
    which clutter up your thinking
    about the object they define.

History

The discussion below is an over­simpli­fication of the history of mathe­matics, which many people have written thick books about. A book relevant to these ideas is Plato’s Ghost, by Jeremy Gray.

Until late in the nineteenth century, functions were usually thought of as defined by formulas (including infinite series). Problems arose in the theory of harmonic analysis which made mathematicians require a more general notion of function. They came up with the concept of function as a set of ordered pairs with the functional property (discussed below), and that understanding revolutionized our understanding of math.

In particular, this definition, along with the use of set theory, enabled abstract math (ahem) to become a cohttp://www.abstractmath.org/MM/MMon tool for understanding math and proving theorems. It is conceivable that some readers may wish it hadn’t. Well, tough.

The modern definition of function given here (which builds on the ordered pairs with functional property definition) came into use beginning in the 1950’s. The modern definition became necessary in algebraic topology and is widely used in many fields today.

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. Of course, the study of algorithms is one of the central topics of modern computing science, so the notion of function-as-formula (updated to function-as-algorithm) has achieved a new importance in recent years.

To state both the definition, we need a preliminary idea.


The functional property

A set $P$ of ordered pairs has the functional property if two pairs in $P$ with the same first coordinate have to have the same second coordinate (which means they are the same pair). In other words, if $(x,a)$ and $(x,b)$ are both in $P$, then $a=b$.

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 (which is just what requirement FP says in the specification). That’s why you can write “$G(x)$” for any $x$ in the domain of $G$ and not be ambiguous.

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. Note that there are two different pairs with the same second coordinate. This is irrelevant to the functional property.
  • 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.
  • The empty set $\emptyset$ has the function property vacuously.

Example: graph of a function defined by a formula


In calculus books, a picture like this one (of part of $y=x^2+2x+5$) is called a graph. Here I use the word “graph” to denote the set of ordered pairs
\[\left\{ (x,{{x}^{2}}+2x+5)\,\mathsf{|}\,x\in \mathbb{R } \right\}\]
which is a mathematical object rather than some ink on a page or pixels on a screen.

The graph of any function studied in beginning calculus has the functional property. For example, the set of ordered pairs above has the functional property because if $x$ is any real number, the formula ${{x}^{2}}+2x+5$ defines a specific real number.

  • if $x = 0$, then ${{x}^{2}}+2x+5=5$, so the pair $(0, 5)$ is an element of the graph of $G$. Each time you plug in $0$ in the formula you get 5.
  • if $x = 1$, then ${{x}^{2}}+2x+5=8$.
  • if $x = -2$, then ${{x}^{2}}+2x+5=5$.

You can measure where the point $\{-2,5\}$ is on the (picture of) the graph and see that it is on the blue curve as it should be. No other pair whose first coordinate is $-2$ is in the graph of $G$, only $(-2, 5)$. That is because when you plug $-2$ into the formula ${{x}^{2}}+2x+5$, you get $5$ and nothing else. Of course, $(0, 5)$ is in the graph, but that does not contradict the functional property. $(0, 5)$ and $(-2, 5)$ have the same second coordinate, but that is OK.



Mathematical definition of function

A function $f$ is a
mathematical structure consisting of the following objects:

  • A set called the domain of $f$, denoted by $\text{dom} f$.
  • A set called the codomain of $f$, denoted by $\text{cod} f$.
  • A set of ordered pairs called the graph of $ f$, with the following properties:
  • $\text{dom} f$ \text{dom} fis the set of all first coordinates of pairs in the graph of $f$.
  • Every second coordinate of a pair in the graph of $f$ is in $\text{cod} f$ (but $\text{cod} f$ may contain other elements).
  • The graph of $f$ has the functional property.

Using arrow notation, this implies that $f:\text{dom}f\to\text{cod} f$.

Remark

The main difference between the specification of function given previously and this definition is that the definition replaces the statement “$f$ has a value at $a$” by introducing a set of ordered pairs (the graph) with the functional property.

  • This set of ordered pairs is extra structure introduced by the definition mainly in order to make the definition a classical sets-with-structure.
  • This makes the graph, which should be a concept derived from the concept of function, appear to be a necessary part of the function.
  • That suggests incorrectly that the graph is more of a primary intuition that other intuitions such as function as map, function as transformer, and other points of view discussed in the article Images and meta­phors for functions.
  • The concept of graph of a function is indeed an important intuition, and is discussed with examples in the articles Graphs of continuous functions and Graphs of finite functions.
  • Nevertheless, the fact that the concept of graph appears in the definition of function does not make it the most important intuition.

Examples

  • Let $F$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$ and define $A = \{1, 2, 3, 5\}$ and $B = \{2, 4, 8\}$. Then $F:A\to B$ is a function. In speaking, we would usually say, “$F$ is a function from $A$ to $B$.”
  • Let $G$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$ (same as above), and define $A = \{1, 2, 3, 5\}$ and $C = \{2, 4, 8, 9, 11, \pi, 3/2\}$. Then $G:A\to C$ is a (admittedly ridiculous) function. Note that all the second coordinates of the graph are in the codomain $C$, along with a bunch of miscellaneous suspicious characters that are not second coordinates of pairs in the graph.
  • Let $H$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$. Then $H:A\to \mathbb{R}$ is a function, since $2$, $4$ and $8$ are all real numbers.
  • Let $D = \{1, 2, 5\}$ and $E = \{1, 2, 3, 4, 5\}$. Then there is no function $D\to A$ and no function $E\to A$ with graph $\{(1,2), (2,4), (3,2), (5,8)\}$. Neither $D$ nor $E$ has exactly the same elements as the first coordinates of the graph.

Identity and inclusion

Suppose we have two sets  A and  B with $A\subseteq B$.

  • The identity function on A is the function ${{\operatorname{id}}_{A}}:A\to A$ defined by ${{\operatorname{id}}_{A}}(x)=x$ for all $x\in A$. (Many authors call it ${{1}_{A}}$).
  • When $A\subseteq B$, the inclusion function from $A$ to $B$ is the function $i:A\to B$ defined by $i(x)=x$ for all $x\in A$. Note that there is a different function for each pair of sets $A$ and $B$ for which $A\subseteq B$. Some authors call it ${{i}_{A,\,B}}$ or $\text{in}{{\text{c}}_{A,\,B}}$.

The identity function and an inclusion function for the same set $A$ have exactly the same graph, namely $\left\{ (a,a)|a\in A \right\}$. More about this below.

Other definitions of function

Original abstract definition of function

Definition

  • A function $f$ is a set of ordered pairs with the functional property.
  • If $f$ is a function according to this definition, the domain of $f$ is the set of first coordinates of all the pairs in $f$.
  • If $x\in \text{dom} f$, then we define the value of $f$ at $x$, denoted by $f(x)$, to be the second coordinate of the only ordered pair in $f$ whose first coordinate is $x$.

Remarks

  • This definition is still widely used in mathematical writing.
  • Many authors do not tell you which definition they are using.
  • For many purposes (including freshman calculus for the most part) it does not matter which definition is used.
  • In some branches of math, the modern definition adds great clarity to many complicated situations; using the older definition can even make it difficult to describe some important constructions. There is more about this in New Approaches below.

Possible confusion

Some confusion can result because of the presence of these two different definitions.

  • For example, since the identity function ${{\operatorname{id}}_{A}}:A\to A$ and the inclusion function ${{i}_{A,\,B}}:A\to B$ have the same graph, users of the older definition are required in theory to say they are the same function.
  • Also it requires you to say that the graph of a function is the same thing as the function.
  • In my observation, this does not make a problem in practice, unless there is a very picky person in the room.
  • It also appears to me that the modern definition is (quite rightly) winning and the original abstract definition is disappearing.

Multivalued function

The phrase multivalued function refers to an object that is like a function $f:S\to T$ except that for $s\in S$, $f(s)$ may denote more than one value.

Examples

  • Multivalued functions arose in considering complex functions. In cohttp://www.abstractmath.org/MM/MMon practice, the symbol $\sqrt{4}$ denoted $2$, although $-2$ is also a square root of $4$. But in complex function theory, the square root function takes on both the values $2$ and $-2$. This is discussed in detail in Wikipedia.
  • The antiderivative is an example of a multivalued operator. For any constant $C$, $\frac{x^3}{3}+C$ is an antiderivative of $x^2$, so that $\frac{x^3}{3}$, $\frac{x^3}{3}+42$, $\frac{x^3}{3}-1$ and $\frac{x^3}{3}+2\pi$ are among the infinitely many antiderivatives of $x^2$.

A multivalued function $f:S\to T$ can be modeled as a function with domain $S$ and codomain the set of all subsets of $T$. The two meanings are equivalent in a strong sense (naturally equivalent). Even so, it seems to me that they represent two differ­ent ways of thinking about
multivalued functions. (“The value may be any of these things…” as opposed to “The value is this whole set of things.”)

Some older mathematical papers in com­plex func­tion theory do not tell you that their functions are multi­valued. There was a time when com­plex func­tion theory was such a Big Deal in research mathe­matics that the phrase “func­tion theory” meant complex func­tion theory and every mathe­ma­tician with a Ph. D. knew that complex functions were multi­valued.

Partial function

A partial function $f:S\to T$ is just like a function except that its input may be defined on only a subset of $S$. For example, the function $f(x):=\frac{1}{x}$ is a partial function from the real numbers to the real numbers.

This models the behavior of computer programs (algorithms): if you consider a program with one input and one output as a function, it may not be defined on some inputs because for them it runs forever (or gives an error message).

In some texts in computing science and mathematical logic, a function is by
convention a partial function, and this fact may not be mentioned explicitly, especially in research papers.

The phrases “multivalued function” and “partial function” upset some picky types who say things like, “But a multi­valued func­tion is not a func­tion!”. A hot dog is not a dog, either. I once had a Russian teacher who was Polish and a German teacher who was Hungarian. So what? See the Hand­book (click on
radial category).

New approaches to functions

All the definitions of function given here produce mathematical structures, using the traditional way to define mathematical objects in terms of sets. Such definitions have disadvantages.

Mathematicians have many ways to think about functions. That a function is a set of ordered pairs with a certain property (functional) and possibly some ancillary ideas (domain, codomain, and others) is not the way we usually think about them$\ldots$Except when we need to reduce the thing we are studying to its absolutely most abstract form to make sure our proofs are correct.
That most abstract form is what I have called the rigorous view or the dry bones and it is when that reasoning is needed that the sets-with-structure approach has succeeded.

Our practice of abstraction has led us to new approaches to talking about functions. The most important one currently is category theory. Roughly, a category is a bunch of objects together with some arrows going between them that can be composed head to tail. Functions between sets are examples of this: the sets are the objects and the functions the arrows. But arrows in a category do not have to be functions; in that way category theory is an abstraction of functions.

This abstracts the idea of function in a way that brings out common ideas in various branches of math. Research papers in many branches of mathematics now routinely use the language of category theory. Categories now appear in some undergraduate math courses, meaning that Someone needs to write a chapter on category theory for abstractmath.org.

Besides category theory, computing scientists have come up with other abstract ways of dealing with functions, for example type theory. It has not come as far along as category theory, but has shown recent signs of major progress.

Both category theory and type theory define math objects in terms of their effect on and relationship with other math objects. This makes it possible to do abstract math entirely without using sets-with-structure as a means of defining concepts.

References

  • Functions in Wikipedia. This is an extensive and mostly well-done description of the use of functions in mathematics.

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Introducing abstract topics

I have been busy for the past several years revising abstractmath.org (abmath). Now I believe, perhaps foolishly, that most of the articles in abmath have reached beta, so now it is time for something new.

For some time I have been considering writing introductions to topics in abstract math, some typically studied by undergraduates and some taken by scientists and engineers. The topics I have in mind to do first include group theory and category theory.

The point of these introductions is to get the student started at the very beginning of the topic, when some students give up in total confusion. They meet and fall off of what I have called the abstraction cliff, which is discussed here and also in my blog posts Very early difficulties and Very early difficulties II.

I may have stolen the phrase “abstraction cliff” from someone else.

Group theory

Group theory sets several traps for beginning students.

Multiplication table

  • A student may balk when a small finite group is defined using a set of letters in a multiplication table.
    “But you didn’t say what the letters are or what the multiplication is?”
  • Such a definition is an abstract definition, in contrast to the definition of “prime”, for example, which is stated in terms of already known entities, namely the integers.
  • The multiplication table of a group tells you exactly what the binary operation is and any set with an operation that makes such a table correct is an example of the group being defined.
  • A student who has no understanding of abstraction is going to be totally lost in this situation. It is quite possible that the professor has never even mentioned the concept of abstract definition. The professor is probably like most successful mathematicians: when they were students, they understood abstraction without having to have it explained, and possibly without even noticing they did so.

Cosets

  • Cosets are a real killer. Some students at this stage are nowhere near thinking of a set as an object or a thing. The concept of applying a binary operation on a pair of sets (or any other mathematical objects with internal structure) is completely foreign to them. Did anyone ever talk to them about mathematical objects?
  • The consequence of this early difficulty is that such a student will find it hard to understand what a quotient group is, and that is one of the major concepts you get early in a group theory course.
  • The conceptual problems with multiplication of cosets is similar to those with pointwise addition of functions. Given two functions $f,g:\mathbb{R}\to\mathbb{R}$, you define $f+g$ to be the function \[(f+g)(x):=f(x)+g(x)\] Along with pointwise multiplication, this makes the space of functions $\mathbb{R}\to\mathbb{R}$ a ring with nice properties.
  • But you have to understand that each element of the ring is a function thought of as a single math object. The values of the function are properties of the function, but they are not elements of the ring. (You can include the real numbers in the ring as constant functions, but don’t confuse me with facts.)
  • Similarly the elements of the quotient group are math objects called cosets. They are not elements of the original group. (To add to the confusion, they are also blocks of a congruence.)

Isomorphic groups

  • Many books, and many professors (including me) regard two isomorphic groups as the same. I remember getting anguished questions: “But the elements of $\mathbb{Z}_2$ are equivalence classes and the elements of the group of permutations of $\{1,2\}$ are functions.”
  • I admit that regarding two isomorphic groups as the same needs to be treated carefully when, unlike $\mathbb{Z}_2$, the group has a nontrivial automorphism group. ($\mathbb{Z}_3$ is “the same as itself” in two different ways.) But you don’t have to bring that up the first time you attack that subject, any more than you have to bring up the fact that the category of sets does not have a set of objects on the first day you define categories.

Category theory

Category theory causes similar troubles. Beginning college math majors don’t usually meet it early. But category theory has begun to be used in other fields, so plenty of computer science students, people dealing with databases, and so on are suddenly trying to understand categories and failing to do so at the very start.

The G&G post A new kind of introduction to category theory constitutes an alpha draft of the first part of an article introducing category theory following the ideas of this post.

Objects and arrows are abstract

  • Every once in a while someone asks a question on Math StackExchange that shows they have no idea that an object of a category need not have elements and that morphisms need not be functions that take elements to elements.
  • One questioner understood that the claim that a morphism need not be a function meant that it might be a multivalued function.

Duality

  • That misunderstanding comes up with duality. The definition of dual category requires turning the arrows around. Even if the original morphism takes elements to elements, the opposite morphism does not have to take elements to elements. In the case of the category of sets, an arrow in $\text{Set}^{op}$ cannot take elements to elements — for example, the opposite of the function $\emptyset\to\{1,2\}$.
  • The fact that there is a concrete category equivalent to $\text{Set}^{op}$ is a red herring. It involves different sets: the function corresponding to the function just mentioned goes from a four-element set to a singleton. But in the category $\text{Set}^{op}$ as defined it is simply an arrow, not a function.

Not understanding how to use definitions

  • Some of the questioners on Math Stack Exchange ask how to prove a statement that is quite simple to prove directly from the definitions of the terms involved, but what they ask and what they are obviously trying to do is to gain an intuition in order to understand why the statement is true. This is backward — the first thing you should do is use the definition (at least in the first few days of a math class — after that you have to use theorems as well!
  • I have discussed this in the blog post Insights into mathematical definitions (which gives references to other longer discussions by math ed people). See also the abmath section Rewrite according to the definitions.

How an introduction to a math topic needs to be written

The following list shows some of the tactics I am thinking of using in the math topic introductions. It is quite likely that I will conclude that some tactics won’t work, and I am sure that tactics I haven’t mentioned here will be used.

  • The introductions should not go very far into the subject. Instead, they should bring an exhaustive and explicit discussion of how to get into the very earliest part of the topic, perhaps the definition, some examples, and a few simple theorems. I doubt that a group theory student who hasn’t mastered abstraction and what proofs are about will ever be ready to learn the Sylow theorems.
  • You can’t do examples and definitions simultaneously, but you can come close by going through an example step by step, checking each part of the definition.
  • There is a real split between students who want the definitions first
    (most of whom don’t have the abstraction problems I am trying to overcome)
    and those who really really think they need examples first (the majority)
    because they don’t understand abstraction.

  • When you introduce an axiom, give an example of how you would prove that some binary operation satisfies the axiom. For example, if the axiom is that every element of a group must have an inverse, right then and there prove that addition on the integers satisfies the axiom and disprove that multiplication on integers satisies it.
  • When the definition uses some undefined math objects, point out immediately with examples that you can’t have any intuition about them except what the axioms give you. (In contrast to definition of division of integers, where you and the student already have intuitions about the objects.)
  • Make explicit the possible problems with abstractmath.org and Gyre&Gimble) will indeed find it difficult to become mathematical researchers — but not impossible!
  • But that is not the point. All college math professors will get people who will go into theoretical computing science, and therefore need to understand category theory, or into particle physics, and need to understand groups, and so on.
  • By being clear at the earliest stages of how mathematicians actually do math, they will produce more people in other fields who actually have some grasp of what is going on with the topics they have studied in math classes, and hopefully will be willing to go back and learn some more math if some type of math rears its head in the theories of their field.
  • Besides, why do you want to alienate huge numbers of people from math, as our way of teaching in the past has done?
  • “Our” means grammar school teachers, high school teachers and college professors.

Acknowledgment

Thanks to Kevin Clift for corrections.

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Representations of functions III

Introduction to this post

I am writing a new abstractmath chapter called Representations of Functions. It will replace some of the material in the chapter Functions: Images, Metaphors and Representations. This post is a draft of the sections on representations of finite functions.

The diagrams in this post were created using the Mathematica Notebook Constructions for cographs and endographs of finite functions.nb.
You can access this notebook if you have Mathematica, which can be bought, but is available for free for faculty and students at many universities, or with Mathematica CDF Player, which is free for anyone and runs on Windows, Mac and Linux.

Like everything in abstractmath.org, the notebooks are covered by a Creative Commons ShareAlike 3.0 License.

Segments posted so far

Graphs of finite functions

When a function is continuous, its graph shows up as a curve in the plane or as a curve or surface in 3D space. When a function is defined on a set without any notion of continuity (for example a finite set), the graph is just a set of ordered pairs and does not tell you much.

A finite function $f:S\to T$ may be represented in these ways:

  • Its graph $\{(s,f(s))|s\in S\}$. This is graph as a mathematical object, not as a drawing or as a directed graph — see graph (two meanings)).
  • A table, rule or two-line notation. (All three of these are based on the same idea, but differ in presentation and are used in different mathematical specialties.)
  • By using labels with arrows between them, arranged in one of two ways:
  • A cograph, in which the domain and the codomain are listed separately.
  • An endograph, in which the elements of the domain and the codomain are all listed together without repetition.

All these techniques can also be used to show finite portions of infinite discrete functions, but that possibility will not be discussed here.

Introductory Example

Let \[\text{f}:\{a,b,c,d,e\}\to\{a,b,c,d\}\] be the function defined by requiring that $f(a)=c$, $f(b)=a$, $f(c)=c$, $f(d)=b$, and $f(e)=d$.

Graph

The graph of $f$ is the set
\[(a,c),(b,a),(c,c),(d,b),(e,d)\]
As with any set, the order in which the pairs are listed is irrelevant. Also, the letters $a$, $b$, $c$, $d$ and $e$ are merely letters. They are not variables.

Table

$\text{f}$ is given by this table:

This sort of table is the format used in databases. For example, a table in a database might show the department each employee of a company works in:

Rule

The rule determined by the finite function $f$ has the form

\[(a\mapsto b,b\mapsto a,c\mapsto c,d\mapsto b,e\mapsto d)\]

Rules are built in to Mathematica and are useful in many situations. In particular, the endographs in this article are created using rules. In Mathematica, however, rules are written like this:

\[(a\to b,b\to a,c\to c,d\to b,e\to d)\]

This is inconsistent with the usual math usage (see barred arrow notation) but on the other hand is easier to enter in Mathematica.

In fact, Mathematica uses very short arrows in their notation for rules, shorter than the ones used for the arrow notation for functions. Those extra short arrows don’t seems to exist in TeX.

Two-line notation

Two-line notation is a kind of horizontal table.

\[\begin{pmatrix} a&b&c&d&e\\c&a&c&b&d\end{pmatrix}\]

The three notations table, rule and two-line do the same thing: If $n$ is in the domain, $f(n)$ is shown adjacent to $n$ — to its right for the table and the rule and below it for the two-line.

Note that in contrast to the table, rule and two-line notation, in a cograph each element of the codomain is shown only once, even if the function is not injective.

Cograph

To make the cograph of a finite function, you list the domain and codomain in separate parallel rows or columns (even if the domain and codomain are the same set), and draw an arrow from each $n$ in the domain to $f(n)$ in the codomain.

This is the cograph for $\text{f}$, represented in columns

and in rows (note that $c$ occurs only once in the codomain)

Pretty ugly, but the cograph for finite functions does have its uses, as for example in the Wikipedia article composition of functions.

In both the two-line notation and in cographs displayed vertically, the function goes down from the domain to the codomain. I guess functions obey the law of gravity.

Rearrange the cograph

There is no expectation that in the cograph $f(n)$ will be adjacent to $n$. But in most cases you can rearrange both the domain and the codomain so that some of the structure of the function is made clearer; for example:

The domain and codomain of a finite function can be rearranged in any way you want because finite functions are not continuous functions. This means that the locations of points $x_1$ and $x_2$ have nothing to do with the locations of $f(x_1)$ and $f(x_2)$: The domain and codomain are discrete.

Endograph

The endograph of a function $f:S\to T$ contains one node labeled $s$ for each $s\in S\cup T$, and an arrow from $s$ to $s’$ if $f(s)=s’$. Below is the endograph for $\text{f}$.

The endograph shows you immediately that $\text{f}$ is not a permutation. You can also see that with whatever letter you start with, you will end up at $c$ and continue looping at $c$ forever. You could have figured this out from the cograph (especially the rearranged cograph above), but it is not immediately obvious in the cograph the way it in the endograph.

There are more examples of endographs below and in the blog post
A tiny step towards killing string-based math. Calculus-type functions can also be shown using endographs and cographs: See Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s, by Martin Flashman, and my blog posts Endographs and cographs of real functions and Demos for graph and cograph of calculus functions.

Example: A permutation

Suppose $p$ is the permutation of the set \[\{0,1,2,3,4,5,6,7,8,9\}\]given in two-line form by
\[\begin{pmatrix} 0&1&2&3&4&5&6&7&8&9\\0&2&1&4&5&3&7&8&9&6\end{pmatrix}\]

Cograph

Endograph

Again, the endograph shows the structure of the function much more clearly than the cograph does.

The endograph consists of four separate parts (called components) not connected with each other. Each part shows that repeated application of the function runs around a kind of loop; such a thing is called a cycle. Every permutation of a finite set consists of disjoint cycles as in this example.

Disjoint cycle notation

Any permutation of a finite set can be represented in disjoint cycle notation: The function $p$ is represented by:

\[(0)(1,2)(3,4,5)(6,7,8,9)\]

Given the disjoint cycle notation, the function can be determined as follows: For a given entry $n$, $p(n)$ is the next entry in the notation, if there is a next entry (instead of a parenthesis). If there is not a next entry, $p(n)$ is the first entry in the cycle that $n$ is in. For example, $p(7)=8$ because $8$ is the next entry after $7$, but $p(5)=3$ because the next symbol after $5$ is a parenthesis and $3$ is the first entry in the same cycle.

The disjoint cycle notation is not unique for a given permutation. All the following notations determine the same function $p$:

\[(0)(1,2)(4,5,3)(6,7,8,9)\]
\[(0)(1,2)(8,9,6,7)(3,4,5)\]
\[(1,2)(3,4,5)(0)(6,7,8,9)\]
\[(2,1)(5,3,4)(9,6,7,8)\]
\[(5,3,4)(1,2)(6,7,8,9)\]

Cycles such as $(0)$ that contain only one element are usually omitted in this notation.

Example: A tree

Below is the endograph of a function \[t:\{0,1,2,3,4,5,6,7,8,9\}\to\{0,1,2,3,4,5,6,7,8,9\}\]

This endograph is a tree. The graph of a function $f$ is a tree if the domain has a particular element $r$ called the root with the properties that

  • $f(r)=r$, and
  • starting at any element of the domain, repreatedly applying $f$ eventually produces $r$.

In the case of $t$, the root is $4$. Note that $t(4)=4$, $t(t(7))=4$, $t(t(t(9)))=4$, $t(1)=4$, and so on.

The endograph

shown here is also a tree.

See the Wikipedia article on trees for the usual definition of tree as a special kind of graph. For reading this article, the definition given in the previous paragraph is sufficient.

The general form of a finite function

This is the endograph of a function $t$ on a $17$-element set:

It has two components. The upper one contains one $2$-cycle, and no matter where you start in that component, when you apply $t$ over and over you wind up flipping back and forth in the $2$-cycle forever. The lower component has a $3$-cycle with a similar property.

This illustrates a general fact about finite functions:

  • The endograph of any finite function contains one or more components $C_1$ through $C_k$.
  • Each component $C_k$ contains exactly one $n_k$ cycle, for some integer $n_k\geq 1$, to which are attached zero or more trees.
  • Each tree in $C_k$ is attached in such a way that its root is on the unique cycle contained in $C_k$.

In the example above, the top component has three trees attached to it, two to $3$ and one to $4$. (This tree does not illustrate the fact that an element of one of the cycles does not have to have any trees attached to it).

You can check your understanding of finite functions by thinking about the following two theorems:

  • A permutation is a finite function with the property that its cycles have no trees attached to them.
  • A tree is a finite function that has exactly one component whose cycle is a $1$-cycle.


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Representations of functions II

Introduction to this post

I am writing a new abstractmath chapter called Representations of Functions. It will replace some of the material in the chapter Functions: Images, Metaphors and Representations.

This post includes a draft of the introduction to the entire new chapter (immediately below) and of the sections on graphs of continuous functions of one variable with values in the plane and in 3-space. Later posts will concern multivariable continuous functions and finite discrete functions.

Introduction to the new Chapter

Functions can be represented visually in many different ways. There is a sharp difference between representing continuous functions and representing discrete functions.

For a continuous function $f$, $f(x)$ and $f(x’)$ tend to be close together when $x$ and $x’$ are close together. That means you can represent the values at an infinite number of points by exhibiting them for a bunch of close-together points. Your brain will automatically interpret the points nearby that are not represented.

Nothing like this works for discrete functions. Many different arrangements of the inputs and outputs can be made. Different arrangements may be useful for representing different properties of the function.

Illustrations

The illustrations were created using these Mathematica Notebooks:

These notebooks contain many more examples of the ways functions can be represented than are given in this article. The notebooks also contain some manipulable diagrams which may help you understand the diagrams. In addition, all the 3D diagrams can be rotated using the cursor to get different viewpoints. You can access these tools if you have Mathematica, which is available for free for faculty and students at many universities, or with Mathematica CDF Player, which runs on Windows, Mac and Linux.

Like everything in abstractmath.org, the notebooks are covered by a Creative Commons ShareAlike 3.0 License.

Segments posted so far

Functions from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}$

Suppose $F:\mathbb{R}\to\mathbb{R}\times\mathbb{R}$. That means you put in one number and get out a pair of numbers.

The unit circle

An example is the unit circle, which is the graph of the function $t\mapsto(\cos t,\sin t)$. That has this parametric plot:

Unit circle

Because $\cos^2 t+\sin^2 t=1$, every real number $t$ produces a point on the unit circle. Four point are shown. For example,\[(\cos\pi,\,\sin\pi)=(-1,0)\] and
\[(\cos(5\pi/3),\,\sin(5\pi/3))=(\frac{1}{2},\frac{\sqrt3}{2})\approx(.5,.866)\]

$t$ as time

In graphing functions $f:\mathbb{R}\to\mathbb{R}$, the plot is in two dimensions and consists of the points $(x,f(x))$: the input and the output. The parametric plot shown above for $t\mapsto(\cos^2 t+\sin^2)$ shows only the output points $(\cos t,\sin t)$; $t$ is not plotted on the graph at all. So the graph is in the plane instead of in three-dimensional space.

An alternative is to use time as the third dimension: If you start at some number $t$ on the real line and continually increase it, the value $f(t)$ moves around the circle counterclockwise, repeating every $2\pi$ times. If you decrease $t$, the value moves clockwise. The animated gif circlemovie.gif shows how the location of a point on the circle moves around the circle as $t$ changes from $0$ to $2\pi$. Every point is traversed an infinite number of times as $t$ runs through all the real numbers.

The unit circle with $t$ made explicit

Since we have access to three dimensions, we can show the input $t$ explicitly by using a three-dimensional graph, shown below. The blue circle is the function $t\mapsto(\cos t,\sin t,0)$ and the gold helix is the function $t\mapsto(\cos t,\sin t,.2t)$.

Unit circle

The introduction of $t$ as the value in the vertical direction changes the circle into a helix. The animated .gif covermovie.gif shows both the travel of a point on the circle and the corresponding point on the helix.

As $t$ changes, the circle is drawn over and over with a period of $2\pi$. Every point on the circle is traversed an infinite number of times as $t$ runs through all the real numbers. But each point on the helix is traversed exactly once. For a given value of $t$, the point on the helix is always directly above or below the point on the circle.

The helix is called the universal covering space of the circle, and the set of points on the helix over (and under) a particular point $p$ on the circle is called the fiber over $p$. The universal cover of a space is a big deal in topology.

Figure-8 graph

This is the parametric graph of the function $t\mapsto(\cos t,\sin 2t)$.

Figure 8

Notice that it crosses itself at the origin, when $t$ is any odd multiple of $\frac{\pi}{2}$.

Below is the universal cover of the Figure-8 graph. As you can see, the different instances of crossing at $(0,0)$ are separated. The animated.gif Fig8movie shows the paths taken as $t$ changes on the figure 8 graph and on its universal cover

Unit circle

Functions from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$

The graph of a function from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ can also be drawn as a parametric graph in three-dimensional space, giving a three-dimensional curve. The trick that I used in the previous section of showing the input parameter so that you can see the universal cover won’t work in this case because it would require four dimensions.

Universal covers

The gold curves in the figures for the universal covers of the circle and the figure 8 are examples of functions from $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$.

The seven-pointed crown

Here are views from three different angles of the graph of the function $t\mapsto(\cos t, \sin t, \sin 7t)$:

The animated gif crownmovie.gif represents the parameter $t$ in time.

Another curve in space

Below are two views of the curve defined by $t\mapsto({-4t^2+53t)/18,t,.4(-t^2+1-10t)}$.

The following plots the $x$-curve $-4t^2+53t)/18$ gold in the $yz$ plane and the $z$ curve $.4(-t^2+1-10t)$ in the $xy$ plane. The first and third views are arranged so that you see the curve just behind one of those two planes.



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Very early difficulties in studying abstract math

Introduction

There are a some difficulties that students have at the very beginning of studying abstract math that are overwhelmingly important, not because they are difficult to explain but because too many teachers don’t even know the difficulties exist, or if they do, they think they are trivial and the students should know better without being told. These difficulties cause too many students to give up on abstract math and drop out of STEM courses altogether.

I spent my entire career in math at Case Western Reserve University. I taught many calculus sections, some courses taken by math majors, and discrete math courses taken mostly by computing science majors. I became aware that some students who may have been A students in calculus essentially fell off a cliff when they had to do the more abstract reasoning involved in discrete math, and in the initial courses in abstract algebra, linear algebra, advanced calculus and logic.

That experience led me to write the Handbook of Mathematical Discourse and to create the website abstractmath.org. Abstractmath.org in particular grew quite large. It does describe some of the major difficulties that caused good students to fall of the abstraction cliff, but also describes many many minor difficulties. The latter are mostly about the peculiarities of the languages of math.

I have observed people’s use of language since I was like four or five years old. Not because I consciously wanted to — I just did. When I was a teenager I would have wanted to be a linguist if I had known what linguistics is.

I will describe one of the major difficulties here (failure to rewrite according to the definition) with an example. I am planning future posts concerning other difficulties that occur specifically at the very beginning of studying abstract math.

Rewrite according to the definition

To prove that a statement
involving some concepts is true,
start by rewriting the statement
using the definitions of the concepts.

Example

Definition

A function $f:S\to T$ is surjective if for any $t\in T$ there is an $s\in S$ for which $f(s)=t$.

Definition

For a function $f:S\to T$, the image of $f$ is the set \[\{t\in T\,|\,\text{there is an }s\in S\text{ for which }f(s)=t\}\]

Theorem

Let $f:S\to T$ be a function between sets. Then $f$ is surjective if and only if the image of $f$ is $T$.

Proof

If $f$ is surjective, then the statement “there is an $s\in S$ for which $f(s)=t$” is true for any $t\in T$ by definition of surjectivity. Therefore, by definition of image, the image of $f$ is $T$.

If the image of $f$ is $T$, then the definition of image means that there is an $s\in S$ for which $f(s)=t$ for any $t\in T$. So by definition of surjective, $f$ is surjective.

“This proof is trivial”

The response of many mathematicians I know is that this proof is trivial and a student who can’t come up with it doesn’t belong in a university math course. I agree that the proof is trivial. I even agree that such a student is not a likely candidate for getting a Ph.D. in math. But:

  • Most math students in an American university are not going to get a Ph.D. in math. They may be going on in some STEM field or to teach high school math.
  • Some courses taken by students who are not math majors take courses in which simple proofs are required (particularly discrete math and linear algebra). Some of these students may simply be interested in math for its own sake!

A sizeable minority of students who are taking a math course requiring proofs need to be told the most elementary facts about how to do proofs. To refuse to explain these facts is a disfavor to the mathematics community and adds to the fear and dislike of math that too many people already have.

These remarks may not apply to students in many countries other than the USA. See When these problems occur.

“This proof does not describe how mathematicians think”

The proof I wrote out above does not describe how I would come up with a proof of the statement, which would go something like this: I do math largely in pictures. I envision the image of $f$ as a kind of highlighted area of the codomain of $f$. If $f$ is surjective, the highlighting covers the whole codomain. That’s what the theorem says. I wouldn’t dream of writing out the proof I gave about just to verify that it is true.

More examples

Abstractmath.org and Gyre&Gimble contain several spelled-out theorems that start by rewriting according to the definition. In these examples one then goes on to use algebraic manipulation or to quote known theorems to put the proof together.

Comments

This post contains testable claims

Herein, I claim that some things are true of students just beginning abstract math. The claims are based largely on my teaching experience and some statements in the math ed literature. These claims are testable.

When these problems occur

In the United States, the problems I describe here occur in the student’s first or second year, in university courses aimed at math majors and other STEM majors. Students typically start university at age 18, and when they start university they may not choose their major until the second year.

In much of the rest of the world, students are more likely to have one more year in a secondary school (sixth form in England lasts two years) or go to a “college” for a year or two before entering a university, and then they get their bachelor’s degree in three years instead of four as in the USA. Not only that, when they do go to university they enter a particular program immediately — math, computing science, etc.

These differences may mean that the abstract math cliff occurs early in a student’s university career in the USA and before the student enters university elsewhere.

In my experience at CWRU, some math majors fall of the cliff, but the percentage of computing science students having trouble was considerably greater. On the other hand, more of them survived the discrete math course when I taught it because the discrete math course contain less abstraction and more computation than the math major courses (except linear algebra, which had a balance similar to the discrete math course — and was taken by a sizeable number of non-math majors).

References

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Functions: Metaphors, Images and Representations

Please read this post at abstractmath.org. I originally posted the document here but some of the diagrams would not render, and I haven’t been able to figure out why. Sorry for having to redirect.

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Demos for graph and cograph of calculus functions

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook GraphCograph.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work properly.

This post provides interactive examples of the endograph and cograph of real functions. Those two concepts were defined and discussed in the previous post Endograph and cograph of real functions.

Such representations of functions, put side by side with the conventional graph, may help students understand how to interpret the usual graph representation. For example: What does it mean when the arrows slant to the left? spread apart? squeeze together? flip over? Going back and forth between the conventional graph and the cograph or engraph for a particular function should make you much more in tune to the possibilities when you see only the conventional graph of another function.

This is not a major advance for calculus teachers, but it may be a useful tool.

Line segment

$y=a x+b$

Cubic

$y=a x^3-b x$

Sine

$y=\sin a x$.

Sine and its derivative

$y=\sin a x$ (blue) and $y=a\cos a x$ (red)

Quintic with three parameters

$y=a x^5-b
x^4-0.21 x^3+0.2 x^2+0.5 x-c$

Thanks to Martin Flashman for corrections.

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Presenting binary operations

This is the first of a set of notes I am writing to help me develop my thoughts about how particular topics in my book Abstracting algebra should be organized. This article describes my plan for the book in some detail. The present post has some thoughts about presenting binary operations.

Before binary operations are introduced

Traditionally, an abstract algebra book assumes that the student is familiar with high school algebra and will then proceed with an observation that such operations as $+$ and $\times$ can be thought of as functions of two variables that take a number to another number. So the first abstract idea is typically the concept of binary operation, although in another post I will consider whether that really should be the first abstract concept.

The Abstracting Algebra book will have a chapter that presents concrete examples of algebraic operations and expressions on numbers as in elementary school and as in high school algebra. This section of the post outlines what should be presented there. Each subsection needs to be expanded with lots of examples.

In elementary school

In elementary school you see expressions such as

  • $3+4$
  • $3\times 4$
  • $3-4$

The student invariably thinks of these expressions as commands to calculate the value given by the expression.

They will also see expressions such as
\[\begin{equation}
\begin{array}[b]{r}
23\\
355\\
+ 96\\
\hline
\end{array}
\end{equation}\]
which they will take as a command to calculate the sum of the whole list:
\[\begin{equation}
\begin{array}[b]{r}
23\\
355\\
+ 96\\
\hline
474
\end{array}
\end{equation}\]

That uses the fact that addition is associative, and the format suggests using the standard school algorithm for adding up lists. You don’t usually see the same format with more than two numbers for multiplication, even though it is associative as well. In some elementary schools in recent years students are learning other ways of doing arithmetic and in particular are encouraged to figure out short cuts for problems that allow them. But the context is always “do it”, not “this represents a number”.

Algebra

In algebra you start using letters for numbers. In algebra, “$a\times b$” and “$a+b$” are expressions in the symbolic language of math, which means they are like noun phrases in English such as “My friend” and “The car I bought last week and immediately totaled” in that both are used semantically as names of objects. English and the symbolic language are both languages, but the symbolic language is not a natural language, nor is it a formal language.

Example

In beginning algebra, we say “$3+5=8$”, which is a (true) statement.

Basic facts about this equation:

The expressions “$3+5$” and “$8$”

  • are not the same expression
  • but in the standard semantics of algebra they have the same meaning
  • and therefore the equation communicates information that neither “$3+5$” nor “$8$” communicate.

Another example is “$3+5=6+2$”.

Facts like this example need to be communicated explicitly before binary operations are introduced formally. The students in a college abstract algebra class probably know the meaning of an equation operationally (subconsciously) but they have never seen it made explicit. See Algebra is a difficult foreign language.

Note

The equation “$3+5=6+2$” is an expression just as much as “$3+5$” and “$6+2$” are. It denotes an object of type “equation”, which is a mathematical object in the same way as numbers are. Most mathematicians do not talk this way, but they should.

Binary operations

Early examples

Consciousness-expanding examples should appear early and often after binary operations are introduced.

Common operations

  • The GCD is a binary operation on the natural numbers. This disturbs some students because it is not written in infix form. It is associative. The GCD can be defined conceptually, but for computation purposes needs (Euclid’s) algorithm. This gives you an early example of conceptual definitions and algorithms.
  • The maximum function is another example of this sort. This is a good place to point out that a binary operation with the “same” definition cen be defined on different sets. The max function on the natural numbers does not have quite the same conceptual definition as the max on the integers.

Extensional definitions

In order to emphasize the arbitrariness of definitions, some random operations on a small finite sets should be given by a multiplication table, on sets of numbers and sets represented by letters of the alphabet. This will elicit the common reaction, “What operation is it?” Hidden behind this question is the fact that you are giving an extensional definition instead of a formula — an algorithm or a combination of familiar operations.

Properties

The associative and commutative properties should be introduced early just for consciousness-raising. Subtraction is not associative or commutative. Rock paper scissors is commutative but not associative. Groups of symmetries are associative but not commutative.

Binary operation as function

The first definition of binary operation should be as a function. For example, “$+$” is a function that takes pairs of numbers to numbers. In other words, $+:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ is a function.

We then abstract from that example and others like it from specific operations to arbitrary functions $\Delta:S\times S\to S$ for arbitrary sets $S$.

This is abstraction twice.

  • First we replace the example operations by an arbitrary operation. such as multiplication, subtraction, GCD and MAX on $\mathbb{Z}$, or something complicated such as \[(x,y)\mapsto 3(xy-1)^2(x^2+xy^3)^3\].
  • Then we replace sets of numbers by arbitrary sets. An example would be the random multiplication on the set $\{1,2,5\}$ given by the table
    \[
    \begin{array}{c|ccc}
    \Delta& 1&2&5\\
    \hline
    1&2&2&1\\
    2&5&2&1\\
    5&2&1&5
    \end{array}
    \]
    This defines a function $\Delta:\{1,2,5\}\times\{1,2,5\}\to\{1,2,5\}$ for which for example $\Delta(2,1)=5$, or $2\Delta 1=5$. This example uses numbers as elements of the set and is good for eliciting the “What operation is it?” question.
  • I will use examples where the elements are letters of the alphabet, as well. That sort of example makes the students think the letters are variables they can substitute for, another confusion to be banished by the wise professor who know the right thing to say to make it clear. (Don’t ask me; I taught algebra for 35 years and I still don’t know the right thing to say.)

It is important to define prefix notation and infix notation right away and to use both of them in examples.

Other representations of binary operations.

The main way of representing binary operations in Abstracting Algebra will be as trees, which I will cover in later posts. Those posts will be much more interesting than this one.

Binary operations in high school and college algebra

  • Some binops are represented in infix notation: “$a+b$”, “$a-b$”, and “$a\times b$”.
  • “$a\times b$” is usually written “$ab$” for letters and with the “$\times$” symbol for numbers.
  • Some binops have idiosyncratic representation: “$a^b$”, “${a}\choose{b}$”.
  • A lot of binops such as GCD and MAX are given as functions of two variables (prefix notation) and their status as binary operations usually goes unmentioned. (That is not necessarily wrong.)
  • The symbol “$(a,b)$” is used to denote the GCD (a binop) and is also used to denote a point in the plane or an open interval, both of which are not strictly binops. They are binary operations in a multisorted algebra (a concept I expect to introduce later in the book.)
  • Some apparent binops are in infix notation but have flaws: In “$a/b$”, the second entry can’t be $0$, and the expression when $a$ and $b$ are integers is often treated as having good forms ($3/4$) and bad forms ($6/8$).

Trees

The chaotic nature of algebraic notation I just described is a stumbling block, but not the primary reason high school algebra is a stumbling block for many students. The big reason it is hard is that the notation requires students to create and hold complicated abstract structures in their head.

Example

This example is a teaser for future posts on using trees to represent binary operations. The tree below shows much more of the structure of a calculation of the area of a rectangle surmounted by a semicircle than the expression

\[A=wh+\frac{1}{2}\left(\pi(\frac{1}{2}w)^2\right)\]
does.

The tree explicitly embodies the thought process that leads to the formula:

  • You need to add the area of the rectangle and the area of the semicircle.
  • The area of the rectangle is width times height.
  • The area of the semicircle is $\frac{1}{2}(\pi r^2)$.
  • In this case, $r=\frac{1}{2}w$.

Any mathematician will extract the same abstract structure from the formula\[A=wh+\frac{1}{2}\left(\pi(\frac{1}{2}w)^2\right)\] This is difficult for students beginning algebra.

References

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Inverse image demo revisited

This post is an update of the post Demonstrating the inverse image of a function.

To manipulate the demos in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. CDF Player works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome.

The code for the demos, with some explanatory remarks, is in the file InverseImage.nb on my ,Mathematica website. That website also includes some other examples as .cdf files.

If the diagrams don’t appear, or appear but show pink diagrams, or if the formulas in the text are too high or too low, refresh the screen.

  • The vertical red interval has the horizontal green interval(s) as inverse image.
  • You can move the sliders back and forth to move to different points on the curve. The sliders control the vertical red interval. $a$ is the lower point of the vertical red line and $b$ is the upper point.
  • As you move the sliders back and forth you will see the inverse image breaking up into a disjoint union in intervals, merging into a single interval, or disappearing entirely.
  • The arrow at the upper right makes it run automatically.
  • If you are using Mathematica, you can enter values into the boxes, but if you are using CDF Player, you can only change the number using the slider or the plus and minus incrementers.

 

This is the graph of $y=x^2-1$.

The graph of $-.5 + .5 x + .2 x^2 – .19 x^3 – .015 x^4 + .01 x^5 $

The graph of the rational function $0.5 x+\frac{1.5 \left(x^4-1\right)}{x^4+1}$

The graph of a straight line whose slope can be changed. You can design demos of other functions with variable parameters.

The graph of the sine function. The other demos were coded using the Mathematica Reduce function to get the inverse image. This one had to be done in an ad hoc way as explained in the InverseImage.nb file.

 

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