This post continues Dysfunctions in doing math I, with some more revisions to the article in abstractmath on dysfunctions.

## Elements

### First Myth

**MYTH**: There are two kinds of mathematical objects: "sets" and "elements".

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 |

#### Examples

- The number $42$ is not a set, but it is an element of the set $\{5,10,41,42,-30\}$.
- The sine function is not a set, but it is an element of the set of all differentiable functions defined on the real numbers.
- The set $\{1,2,5\}$
*is*a set, but it is also an element of the set $\left\{\{1,2,5\},\{3,5\}, \emptyset,\{42\}\right\}$. It is*not*an element of the set $\{1,2,3,4,5\}$.

If you find these examples confusing, read this.

### Second Myth

**MYTH**: The empty set is an element of every set.

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.

#### Examples

- The empty set is not an element of every set. It is not an element of the set $\{2,3\}$ for example; that set has only the elements $2$ and $3$.
- The empty set
*is*an element of the set $\{2,3,\emptyset\}$. - The empty set is a subset of every set.

### Other ways to misunderstand sets

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

## Enthymeme

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

#### Example

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.”

- This is an enthymeme; it omits stating, much less proving, that the intersection of transitive relations is transitive.
- The student may “know” that it is obvious that the intersection of transitive relations is transitive, having never considered the similar question of the union of transitive relations.
- It is very possible that the student possesses (probably subconsciously) a malrule to the effect that for any property $P$ the union or intersection of relations with property $P$ also has property $P$.
- The instructor very possibly suspects this. For some students, of course, the suspicion will be unjustified, but for which ones?
- This sort of thing is a frequent source of tension between student and instructor: “Why did you take points off because I assumed the intersection of transitive relations is transitive? It’s
*true*!”

## Malrule

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

#### Example

The malrule $\sqrt{x+y}=\sqrt{x}+\sqrt{y}$ invented by algebra students may come from the pattern given by the distributive law $a(x+y)=ax+ay$. The malrule invented by many first year calculus students that transforms $\frac{d(uv)}{dx}$ to $\frac{du}{dx}\frac{dv}{dx}$ may have been generated by extrapolating from the correct rule

\[\frac{d(u+v)}{dx}=\frac{du}{dx}+\frac{dv}{dx}\] by changing addition to multiplication. Both are examples of “every operation is linear”, which students want desperately to be true, although they are not aware of it.

## Existential bigamy

Beginning abstract math students sometimes make a particular 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.*

#### Example

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 $p = q’m…”$