This post concludes the work begun in Dysfunctions in doing math I and Dysfunctions in doing math II, with more revisions to the article in abstractmath on dysfunctions.
False symmetry
Bases of vector spaces
In a finite dimensional vector space $V$ with subspace $W$, every basis of $W$ can be extended to a basis of $V$. But in general there are bases of $V$ that do not contain a subset that is a basis of $W$. A tragic lack of symmetry that causes innocent students to lose points in linear algebra.
Example
The plane $P$ defined by $x=y$ is a twodimensional 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 $\{(1,1,0),(0,0,1),(0,1,0)\}$ of $\mathbb{R}^3$. But 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
Every subgroup $B$ of a commutative group $A$ is a normal subgroup of $A$. But if $B$ is an commutative subgroup of a noncommutative group $S$, then $B$ may not be a normal subgroup of $S$. For example, $\text{Sym}_3$ (the group of symmetries of an equilateral triangle) has three subgroup 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).
Example
The Fibonacci numbers (MW, Wi) are usually defined inductively like this:
\[F(n)=\left\{ \begin{align}
& 0\text{ if }n=0 \\
& 1\text{ if }n=1 \\
& F(n1)+F(n2)\text{ if }n\gt 1 \\
\end{align} \right.\]
In calculating a sum of Fibonacci numbers, you might write
\[\sum_{k=0}^{n}{F(k)=}\sum_{k=0}^{n}{F(k1)+}\sum_{k=0}^{n}{F(k2)}\]
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
\[\sum_{k=0}^{n}{F(k)=}\sum_{k=1}^{n}{F(k1)+}\sum_{k=2}^{n}{F(k2)}\,\,\,\,\,\,\,\,\,\text{
}(n>1)\]
(The notation “$(n \gt 1)$” means “for all $n$ greater than $1$.” See here )
Literalism
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. The most common form this takes is 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 mathematician would ever think of saying such things.
What is important about an ordered pair is its specification:

An ordered pair has a first coordinate and a second coordinate.
 What the first and second coordinates are completely determine the ordered pair.
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 (that is done here).
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 has a total structure which can be described either by starting with a relation and imposing axioms, or by giving a set of subsets and imposing axioms. Each set of axioms describes exactly the same structure; every theorem that can be deduced from the axioms for an equivalence relation can be deduced from the axioms for a partition.
Functions
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 “$\left( 3,\,7 \right)\in F$”.
Vanishing
Many years ago 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 $x^21$ 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 function’s value at $1$ is $0$. The function is not zero anywhere, it is $x^21$, or if you have other literalness attitudes, it is “the function $f(x)$ defined by $f(x)=x^21$”.
 Even its value doesn’t literally “vanish”. The value is written as “$0$”. Look at it closely. You can see it. It has not vanished.
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.
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.
Example
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.

Example
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 dihedral group of order 8 and the quaternion group 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.
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!
Math ed people have referred to this as the “studentprofessor problem”. But it is not the real studentprofessor problem.
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.
Example
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 $2\cdot 11\cdot49$?
Unique
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 seemed 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.
Avoid rewording 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
Examples
 The statement “Either $x \gt 0$ or $x \lt 2$” is true (for real numbers). Yes, you could make a stronger statement, for example “Either $x\le 0$ or $x \gt 0$”. But the statement “Either $x \gt 0$ or $x \lt 2$” is still true.
 Some students have problems with the true statements “$2\le 2$” and with “$2\le 3$” for a similar reason, since in fact $2 = 2$ and $2 \lt 3$.
 You may get a twinge if someone says “Many primes are odd”, since in fact there is only one that is not
odd. But it is still true that many primes are odd.
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.
There is another example here.
It is not wrong for an author to make an unnecessarily weak assertion.

Rabbits
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:

(*) For any positive real number $\varepsilon $,
 there is a positive real number $\delta $ for which:
 if $\left x2 \right\lt\delta$ then
 $\left x^24 \right\lt\varepsilon$.
Proof
Here is the proof, with what I imagine might be your agitated reaction to certain steps. Below is a proof with detailed explanations .
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 $\left x2 \right\lt\delta$.
4) Then $\left x2 \right\lt1$ by (2) and (3).
5) By (4) and algebra, $\leftx+2 \right\lt5$.
Well, so what? We know that $\left x+39
\right\lt42$ and lots of other things, too. Why did they do this?
6) Also $\left x2 \right\lt\frac{\varepsilon }{5}$ by (2).
7) Then $\left {{x}^{2}}4
\right=\left (x2)(x+2) \right\lt\frac{\varepsilon }{5}\cdot 5=\varepsilon$ by (5) and (6). End of Proof.
Remarks
This proof is typical of proofs in texts.
 Steps 2) and 5) look like they were rabbits pulled out of a hat.
 The author gives no explanation of where they came from.
 Even so, each step of the proof follows from previous steps, so the proof is correct.
 Whether you are surprised or not has nothing to do with whether it is correct.
 In order to understand a proof, you do not have to know where the rabbits came from.
 In general, the author did not think up the proof steps in the order they occur in the proof. (See this remark in the section on Forms of Proofs.)
 See also look ahead.
Acknowledgments
Thanks to Robert Burns for corrections and suggestions
This work is licensed under a Creative Commons AttributionShareAlike 2.5 License.