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.
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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 $p = q’m…”$
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