Tag Archives: multiplication table

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 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.


  • 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.


Thanks to Kevin Clift for corrections.

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Presenting binops as trees

Binary operations as trees

This is one of a series of posts I am writing to help me develop my thoughts about how particular topics in my book Abstracting Algebra (“AbAl“) should be organized. In some parts, I present various options that I have not decided between.

This post concerns the presen­ta­tion of binary operations as trees. The Mathematica code for the diagrams is in Substitution in algebra.nb

Binary operations as functions

A binary operation or binop $\Delta$ is a function of two variables whose value at $(a,b)$ is traditionally denoted by $a\Delta b$. Most commonly, the function is restricted to having inputs and outputs in the same set. In other words, a binary operation is a function $\Delta:S\times S\to S$ defined on some set $S$. $S$ is the underlying set of the operation. For now, this will be the definition, although binops may be generalized to multiple sets later in the book.

In AbAl:

  • Binops will be defined as functions in the way just described.
  • Algebraic expressions will be represented
    as trees, which exhibit more clearly the structure of the expressions that is encoded in algebraic notation.
  • They will also be represented using the usual infix expressions such as “$3\times 5$” and “$3-5$”,

Fine points

The definition of a binop as a function has termi­no­logical consequences. The correct point of view concerning a function is that it determines its domain and its codomain. In particular:

A binary operation determines its underlying set.

Thus if we talk about an arbitrary binop $\Delta$, we don’t have to give a name to its underlying set. We can just say “the underlying set of $\Delta$” or “$U(\Delta)$”.


“$+$” is not one binary operation.

  • $+:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ is a binary operation.
  • $+:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is another binary operation.

Mathematicians commonly refer to these particular binops as “addition on the integers” and “addition on the reals”.


You almost never see this attitude in textbooks on algebra. It is required by both category theory and type theory, two Waves flooding into math. Category theory is a middle-aged Wave and type theory, in the version of homo­topy type theory, is a brand new baby Wave. Both Waves have changed and will change our under­standing of math in deep ways.


An arbitrary binop $\Delta$ can be represented as a binary tree in this way:

generic binop

This tree represents the expression that in standard algebraic notation is “$a\Delta b$”.

In more detail, the tree is an ordered rooted binary tree. The “ordered” part means that the leaves (nodes with no descendants) are in a specific left to right order. In AbAl, I will define trees in some detail, with lots of pictures.

The root shows the operation and the two leaves show elements of the underlying set. I follow the custom in computing science to put the root at the top.

Metaphors should not dictate your life by being taken literally.


The Wikipedia treatment of trees is scat­tered over many articles and they almost always describe things mostly in words, not pictures. Describing math objects in words when you could use pictures is against my religion. Describing is not the same as defining, which usually requires words.

Some concrete examples:



These are represen­ta­tions of the expressions “$3+5$”, “$3\times5$”, and “$3-5$”.

Just as “$5+3$” is a different expression from “$3+5$”, the left tree in 3trees above is a different expression from this one:



They have the same value, but they are distinct as expressions — otherwise, how could you state the commutative law?

Fine points

I regard an expression as an abstract math object that can have many repre­sentations. For example “$3+5$” and the left tree in 3trees are two different represen­ta­tions of the same (abstract) expression. This deviates from the usual idea that “expression” refers to a typographical construction.

In previous posts, when the operation is not commutative, I have sometimes labeled the legs like this:

I have thought about using this notation consistently in AbAl, but I suspect it would be awkward in places.

Evaluation and substitution

The two basic operations on algebraic expressions
are evaluation and substitution.

They and the Only Axiom of Algebra, which I will discuss in a later post, are all that is needed to express the true nature of algebra.


  • If you evaluate $3+5$ you get $8$.
  • If you evaluate $3\times 5$ you get $15$.
  • If you evaluate $3-5$ you get $-2$.

I will show evaluation on trees like this:

Evaluation with trace

A more elaborate version, valuation with trace, would look like this. This allows you to keep track of where the valuations come from.

You could also keep track of the operation used at each node. An interactive illustration of this is in the post Visible algebra I supplement. That illustration requires CDF Player to be installed on your computer. You can get it free from the Mathematica website.


In the tree above, the $a$ and $b$ are variables, just as they are in the equivalent expression $a\Delta b$. Algebra beginners have a hard time understanding variables.

  • You can’t evaluate an expression until you substitute numbers for the letters, which produces an instance of expression. (“Instance” is the preferable name for this, but I often refer to such a thing as an “example”.)
  • If a variable is repeated you have to substitute the same value for each occurrence. So $a\Delta b$ is a different expression from $a\Delta a$: $2+3$ is an instance of $a+b$ but it is not an instance of $a+a$. But $a\Delta a$ and $b\Delta b$ are the same expression: any instance of one is an instance of the other.
  • Substitute $a\Delta b$ for $a$ in $a\Delta b$ and you get $(a\Delta b)\Delta b$. You may have committed variable clash. You might have meant $(a\Delta b)\Delta c$. (Somebody please tell me a good link that describes variable clash.)
  • Later, you will deal with multiplication tables for algebraic structures. There the elements are denoted by letters of the alphabet. They can’t be substituted for.

Empty boxes

A straightforward way to denote variables would be to use empty boxes:

The idea is that a number (element of the underlying set) can be inserted in each box. If $3$ (left) and $5$ (right) are placed in the boxes, evaluation would place the value of $3\Delta5$ in the root. Each empty box represents a separate variable.

Empty boxes could also be used in the standard algebraic notation: $\Delta$ or $+$ or $-$.
I have seen that notation in texts explaining variables, but I don’t know a reference. I expect to use this notation with trees in AbAl.

To achieve the effect of one variable in two different places, as in

we can cause it to repeat, as below, where “$\text{id}$” denotes the identity function on the underlying set:

To evaluate at a number (member of the underlying set) you insert a number into the only empty box

which evaluates to

which of course evaluates to $3\Delta3$.

This way of treating repeated variables exhibits the nature of repeated variables explicitly and naturally, putting the values automatically in the correct places. This process, like everything in this section, comes from monad theory. It also reminds me of linear logic in that it shows that if you want to use a value more than once you have to copy it.


Given two binary trees


you could attach the root of the first one to one of the leaves of the second one, in two different ways, to get these trees:



which in standard algebra notation would be written $(a-b)-c)$ and $a-(b-c)$ respectively. Note that this tree

would be represented in algebra as $(a-b)-b$.

In general, substituting a tree for an input (variable or empty box) consists of replacing the empty box by the whole tree, identifying the root of the new tree with the empty box. In graph theorem, “substitution” may be called “grafting”, which is a good metaphor.

You can evaluate the left tree in 2trees at particular numbers to evaluate it in two stages:

Of course, evaluating the right one at the same values would give you a different answer, since subtraction is not associative. Here is another example:

Binary trees in general

By repeated substitution, you can create general binary trees built up of individual trees of this form:

In AbAl I will give examples of such things and their counterparts in algebraic notation. This will include binary trees involving more than one binop, as well. I showed an example in the previous post, which example I repeat here:

It represents the precise unsimplified expression


Some of the operations in that tree are associative and commutative, which is why the expression can be simplified. The collection of all (finite) binary trees built out of a single binop with no assumption that it satisfies laws (associative, commutative and so on) is the free algebra on that binary operation. It is the mother of all binary operations, so it plays the same role for an arbitrary binop that the set of lists plays for associative operations, as described in Monads for High School III: Algebras. All this will be covered in later chapters of AbAl.


Preceding posts in this series

Other references

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