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Produced by Charles Wells     Revised 2017-02-07
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Boundary values of definitions




Existential bigamy

False symmetry

Jump the fence



Method addiction

Proof by Example


Reading variable names as labels

The representation is the object


Unnecessarily weak assertion


It has been said that movie stars are the royalty of America. The better analogy, really, is that the royals are the movie stars of Britain. —Neal Gabler

An analogy can be helpful, but it isn’t the same thing as the same thing. --The Economist

Analogy may suggest new theorems or ways of doing things. You may think that some kind of behavior of a certain type of math object may also work for another similar kind of object. Sometimes that works out and you get a new theorem.

But analogy is fallible. What happens particularly often with people learning abstract math is applying a rule to a situation where it is not appropriate, for example when in two different cases an operation has the same form.

Matrix multiplication

Matrix multiplication is not commutative

If $r$ and $s$ are real numbers then the products $rs$ and $sr$ are always the same number. In other words, multiplication of real numbers is commutative: $rs = sr$ for all real numbers $r$ and $s$.

The product of two matrices $M$ and $N$ is written $MN$, just as for numbers. But matrix multiplication is not commutative. For example, \[\left( \begin{array}{cc} 1 & 2 \\ 3 & 4\\ \end{array} \right) \left( \begin{array}{cc} 3 & 1 \\ 3 &2\\ \end{array} \right) = \left( \begin{array}{cc} 9 & 5\\ 21 & 11 \\ \end{array} \right)\] but \[\left( \begin{array}{cc} 3 & 1 \\ 3 & 2\\ \end{array} \right) \left(\begin{array}{cc} 1 & 2 \\ 3 & 4\\ \end{array} \right) = \left( \begin{array}{cc} 6 & 10\\ 9 & 14 \\ \end{array} \right)\] Because $rs = sr$ for numbers, the formal similarity of the notation suggests $MN$ = $NM$, which is wrong.

This means, for example, you can't blindly manipulate $MNM$ to become $M^2N$. More generally, a law such as $(MN)^n=M^nN^n$ is not correct when $M$ and $N$ are matrices.

You must know the types
of the symbols you manipulate.

When William Rowan Hamilton was trying to understand the new type of number called quaternions (MW, Wik) that he invented, he assumed by analogy that like other numbers, quaternion multiplication was commutative. It took him a long time to realize that that could not be true. If matrix multiplication had already been well-known when he was struggling with quaternions, he probably would have thought of noncommu­ta­tivity much more quickly.

You could imagine a lightbulb over his head, but lightbulbs hadn't been invented yet.

Canceling sine

This happens occasionally in freshman calculus classes.

Boundary values of definitions

Definitions are usually inclusive

Definitions of math concepts usually include the special cases they generalize.


Definitions usually include trivial and improper cases


But definitions don't always include extreme or limiting cases


A definition that includes trivial or extreme cases may be called inclusive; otherwise it is exclusive. People new to abstract math very commonly use words defined inclusively as if their definition was exclusive.

Vacuous truth

A conditional assertion of the form "If $P$ then $Q$" is true if $P$ is always false. This is an extreme case of of conditional assertions and it causes students no end of grief. It is discussed in detail in the chapter on Conditional Assertions.


There are two possibilities:

Another possibility, which can overlap with the two above, is:

This is not like the "co" in category theory: "product" and "coproduct" have a symmetric relationship with each other, but the co-intimidator relation is asymmetric.

There are many ways to get around being intimidated.

If you don't know something
usually some other students don't know it either.

This statement is based on surveys math educators have made.


Students have several myths about sets and their elements, and some students quote these myths to each other as if they knew what they were talking about.

First Myth

MYTH: There are two kinds of mathematical objects: "sets" and "elements". This is a false statement.

This is the TRUTH: Being an element is not a property that some math objects have and others don't. "Element" is a binary relation; it relates an object and a set. So "$3$ is an element" means nothing, but "$3$ is an element of the set of integers" is true and relates two mathematical objects to each other.

Any mathematical object can be an element of a set
In particular, any set can be the element of another set.


If you find these examples confusing, read this.

Second Myth

MYTH: The empty set is an element of every set. This is a false statement.

This is the TRUTH: The empty set is an element of a set $S$ if and only if the definition of $S$ requires it to be an element.


Other ways to misunderstand sets

The myths just listed are explicit; students tell them to each other. The links below tell you about other misunderstanding about sets which are usually subconscious.


An enthymeme is an argument based partly on unexpressed beliefs. Beginners at the art of writing proofs often produce enthymemes.


In the process of showing that the intersection of two equivalence relations $E$ and $E'$ is also an equivalence relation, a student may write

$E\cap E'$ is transitive because $E$ and $E'$ are transitive."

Existential bigamy

Beginning abstract math students sometimes make a type of mistake that occurs in connection with a property $P$ of an mathe­matical object $x$ that is defined by requiring the existence of an item $y$ with a certain relationship to $x$: When students have a proof that assumes that there are two items $x$ and $x'$ with property $P$, they sometimes assume that the same $y$ serves for both of them.

This mistake is called existential bigamy: The fact that Muriel and Bertha are both married (there is a person to whom Muriel is married and there is a person to whom Bertha is married) doesn't mean they are married to the same person.


Let $m$ and $n$ be integers. By definition, $m$ divides $n$ if there is an integer $q$ such that $n=qm$. Suppose you are asked to prove that if $m$ divides both $n$ and $p$, then $m$ divides $n+p$. If you begin the proof by saying, "Let $n = qm$ and $p = qm$..." then you are committing existential bigamy. You need to begin the proof this way: "Let $n = qm$ and $p = q'm…"$

False symmetry

If you have a statement referring to several objects, you may be able to make another statement by permuting some of the object. Unfortu­nately, the fact that the first statement is true does not mean the second one is true.

Bases of vector spaces

Suppose $V$ is a vector space with subspace $W$. Then this statement is correct:

If $B$ is a basis for $W$, there is a basis $B'$ of $V$ for which $B\subseteq B'$.

Unfortunately, the statement below is in general incorrect:

If $B''$ is a basis for $V$, then $B''$ contains a basis for $W$.

In other words, in general, there are bases of $V$ that do not contain a basis of $W$ as a subset.

This is a tragic lack of symmetry that causes innocent students to lose points in linear algebra courses. If you are teaching linear algebra, you should point this out to the students, even though you will thereby ruin their day.


The plane $P$ defined by $x=y$ is a two-dimensional subspace of the three dimensional Euclidean space with axes $x,y,z$. One basis of $P$ is \[\{(1,1,0),(0,0,1)\}\] It can be extended to the basis \[\{(1,1,0),(0,1,0),(0,0,1)\}\] of $\mathbb{R}^3$. On the other hand, the basis \[\{(1,0,0),(0,1,0),(0,0,1)\}\] of $\mathbb{R}^3$ does not contain a subset that is a basis of $P$.

Normal subgroups


$\text{Sym}_3$ (the group of symmetries of an equilateral triangle) has three subgroups with two elements each. Each subgroup is commutative, but is not a normal subgroup of $\text{Sym}_3$.

Jump the fence

If you are working with an expression whose variables are constrained to certain values, and you substitute a value in the expression that violates the constraint, you jump the fence (also called a fencepost error).


The Fibonacci numbers (MW, Wi) are usually defined inductively like this: \[F(n)=\left\{ \begin{align} & 0\text{ if }n=0 \\ & 1\text{ if }n=1 \\ & F(n-1)+F(n-2)\text{ if }n\gt 1 \\ \end{align} \right.\] In calculating a sum of Fibonacci numbers, you might write \[\sum_{k=0}^{n}{F(k)=}\sum_{k=0}^{n}{F(k-1)+}\sum_{k=0}^{n}{F(k-2)}\] This contains errors: the sums on the right involve $F(-1)$ and $F(-2)$, which are not defined by the definition above. You could add \[F(n)=0\text{ if }n\lt 0\] to the definition to get around this, or keep better track of the fence by writing

\[\sum_{k=0}^{n}{F(k)=}\sum_{k=1}^{n}{F(k-1)+}\sum_{k=2}^{n}{F(k-2)}\,\,\,\,\,\,\,\,\,\text{ }(n>1)\]

(The notation "$(n \gt 1)$" means "for all $n$ greater than $1$." See here )


Every type of math object has to have a definition. In giving a definition, a few of the many ingredients that are involved in that type of object are selected as a basis for the definition. They are not necessarily the most important parts. People who make definitions try to use as little as possible in the definition so that it is easier to verify that something is an example of the thing being defined.

A definitional literalist is someone who insists on thinking about a type of math object primarily in terms of what the definition says it is.

Definitional literalism inhibits
your understanding of abstract math.

Ordered pairs

One of the major tools in the study of the foundations of mathematics is to try to define all mathematical objects in terms of as few as possible objects. Until the advent of category theory, the most common form this took was to define everything in terms of sets. For example, the ordered pair $(a,b)$ can be defined to be the set $\{a, \{a, b\}\}$. (See Wi). A definitional literalist will conclude that the ordered pair $(a,b)$ is the set $\{a, \{a, b\}\}$.

This would mean that it makes sense to say that $a\in(a,b)$ but $b\notin(a,b)$. No reasonable mathematician would ever think of saying such things.

What is important about an ordered pair is its specification:

It is ludicrous to say something like "$a\in (a,b)$". The "definition" that $(a,b)$ is the set $\{a,\{\{a,b\}\}$ is done purely for the purpose of showing that the study of ordered pairs can be reduced to the study of sets. It is not a fact about ordered pairs that we can use.

Equivalence relations

An equivalence relation on a set S is a relation on S with certain properties. A partition on S is a set of subsets with certain properties. The two definitions can be proven to give the same structure (see Wikipedia).

I have personally heard literalists say, "How can they give the same structure? One is a relation and one is a partition." The point is that an equivalence relation/partition on a set has a total structure which can be described by axioms in two ways:

Each set of axioms describes exactly the same total structure. Every theorem that can be deduced from the axioms for the equivalence relation can be deduced from the axioms for the partition, and vice versa.


The (less strict) definition of function says that a function is a set of ordered pairs with the functional property.

This does not mean that if your function is $F ( x ) = 2 x + 1$, then you would say "$\left( 3,\,7 \right)\in F$" . The most common practice is to say that "$F (3) = 7$" or "the value of $F$ at $3$ is $7$" or something of the sort.

I do know mathematicians who tell me that they really do think of a function as a set of ordered pairs and would indeed say "$\left( 3,\,7 \right)\in F$".


When I was a graduate student, I had a math professor who hated it with a purple passion if anyone said a function vanishes at some number $a$, meaning its value at $a$ is $0$. If you said, "The function $x^2-1$ vanishes at $1$", he would say, "Pah! The function is still there isn't it?"

There are in fact two different points a literalist can make about such a statement.

The phrase "the function vanishes at $a$" is a metaphor. Mathematicians use metaphors in writing and talking about math all the time, just as people do in writing and talking about anything. Nevertheless, being occasionally the obnoxious literalist sometimes clears up misunderstanding. That is why mathematicians have a reputation for literalism.


A malrule is an incorrect rule for syntactic transformation of a mathematical expression.


Both are examples of the universal malrule "every operation is linear", which many students unconsciously expect to be true, although they are not aware of it.

Method addiction

Beginners at abstract math sometimes have the attitudes that a problem must be solved or a proof constructed by a specific procedure. They become quite uncomfortable when faced with problem solutions that involve guessing or conceptual proofs that involve little or no calculation.


Once I gave a problem in my Theoretical Computer Science class that in order to solve it required finding the largest integer $n$ for which $n!\lt109$. Most students solved it correctly, but several wrote apologies on their paper for doing it by trial and error. Of course:

Trial and error is a perfectly valid method.


Students at a more advanced level may feel insecure in the case where they are faced with solving a problem for which they know there is no known feasible algorithm, a situation that occurs mostly in senior and graduate level classes. For example, there are no known feasible general algorithms for determining if two finite groups given by their multiplication tables are isomorphic, and there is no algorithm at all to determine if two presentations (generators and relations) give the same group.

Even so, the question, "Are the quaternion group and the group of symmetries of the square isomorphic?" is not hard. (Answer: No, they have different numbers of elements of order $2$ and $4$.)

Sometimes you can solve special cases of unsolvable problems.

See also look ahead and conceptual.

Proof by Example

Definition: An integer is even if it is divisible by 2.

Theorem : Prove that if $n$ is an even integer then so is ${{n}^{2}}$.

This is proved by universal generalization .

One type of mistake made by beginners for proofs like this would be the following:

"Proof: Let $n = 8$. Then ${{n}^{2}}=64$ and $64$ is even."

This violates the requirement of universal generalization that you have "made no restrictions on $c$" – you have restricted it to being a particular even integer!

It may be that some people who make this kind of mistake don't understand universal generalization (see also bound variable). But for others, the mistake is caused by misreading the phrase "An integer is even if…" to read that you can prove the statement by picking an integer and showing that it is true for that integer. But in fact, "an" in a statement like this means "any". See indefinite article.


Sometimes when you are reading or listening to a proof you will find yourself following each step but with no idea why these steps are going to give a proof. This can happen with the whole structure of the proof or with the sudden appearance of a step that seems like the prover pulled a rabbit out of a hat . You feel as if you are walking blindfolded.

Example (mysterious proof structure)

The lecturer says he will prove that for an integer $n$, if $n^2$ is even then $n$ is even. He begins the proof: Let $n^2$ be odd" and then continues to the conclusion, "Therefore $n$ is odd."

Why did he begin a proof about being even with the assumption that $n$ is odd?

The answer is that in this case he is doing a proof by contrapositive . If you don't recognize the pattern of the proof you may be totally lost. This can happen if you don't recognize other forms, for example contradiction and induction.

Example (rabbit)

You are reading a proof that $\underset{x\to 2}{\mathop{\lim }}{{x}^{2}}=4$. It is an $\varepsilon \text{-}\delta$ proof, so what must be proved is:


Here is the proof, with what I imagine might be your agitated reaction to certain steps.

1) Suppose $\varepsilon \gt0$ is given.

2) Let $\delta =\text{min}\,(1,\,\frac{\varepsilon }{5})$ (the minimum of the two numbers 1 and $\frac{\varepsilon}{5}$ ).

Where the *!#@! did that come from? They pulled it out of thin air! I can't see where we are going with this proof!

3) Suppose that $\left| x-2 \right|\lt\delta$.

4) Then $\left| x-2 \right|\lt1$ by (2) and (3).

5) By (4) and algebra, $\left|x+2 \right|\lt5$.

Well, so what? We know that $\left| x+39 \right|\lt42$ and lots of other things, too. Why did they do this?

6) Also $\left| x-2 \right|\lt\frac{\varepsilon }{5}$ by (2).

7) Then $\left| {{x}^{2}}-4 \right|=\left| (x-2)(x+2) \right|\lt\frac{\varepsilon }{5}\cdot 5=\varepsilon$ by (5) and (6). End of Proof.


This proof is typical of proofs in texts.


Thanks to Robert Burns for corrections and suggestions

Reading variable names as labels

An assertion such as “There are six times as many students as professors” is translated by some students as $6s = p$ instead of $6p = s$ (where $p$ and $s$ have the obvious meanings). This sort of thing can be avoided by plugging in numbers for the variables to see if the resulting equations make sense. You know it's wrong to say that if you have $12$ professors then you have $2$ students! This problem is discussed in more detail here.

The representation is the object

Many newbies at abstract mathematics firmly believe that the number $735$ is the expression “735”. In fact, the number $735$ is an abstract math object, not a string of symbols that represents the number. This attitude inhibits your ability to use whatever representation of an object is best for the purpose.


Someone faced with a question such as “Does $21$ divide $3 \cdot5\cdot72$?” may immediately multiply the expression out to get $1080$ and then carry out long division to see if indeed $21$ divides $1080$. They will say things such as, “I can’t tell what the number is until I multiply it out.”

In this example, it is easy to see that $21$ does not divide $3 \cdot5\cdot72$, because if it did, $7$ would be a prime factor, but $7$ does not divide $72$.

Integers have many representations: decimal, binary, the prime factorization, and so on. Clearly the prime factorization is the best form for determining divisors, whereas for example the decimal notation is a good form for determining which of two integers is the larger. For example, is $3 \cdot5\cdot72$ bigger or smaller than $2\cdot 11\cdot49$?


By definition, a set $R$ of ordered pairs has the functional property if two pairs in $R$ with the same first coordinate have to have the same second coordinate

It is wrong to rephrase the definition this way: "The first coordinate determines a unique second coordinate." That use of "unique" is ambiguous. It could mean the set \[\{(1,2), (2,4), (3,2), (5,8)\}\] does not have the functional property because the first coordinate in $(1,2)$ determines $2$ and the first coordinate in $(3,2)$ determines $2$, so it is "not unique". This statement is wrong. The set does have the functional property.

A related error is to reword the definition of injective by saying, "For each input there is a unique output." It is easy to read this and think injectivity is merely the functional property.

It seems to me that during the 35 years I taught calculus and discrete math, students fell into this trap about 100,000 times. Of course, this could be a slight exaggeration.

DO NOT reword any definition that does not use the word unique
so that it DOES use the word unique.
Such activity fries your brain and turns A's into B's.

Unnecessarily weak assertion


An unnecessarily weak assertion may occur in math texts because it is the form your proof gives you, or it is the form you need for a proof. In the latter case you may feel the author has pulled a rabbit out of a hat.

It is not wrong for an author to make an unnecessarily weak assertion.

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