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I had the good fortune of assigning the following problem on the first day of my freshman honors calculus course:

Is $a$ contained in the interval $[a,a)$? Explain.

The majority of my students wrote incorrect responses. A representative of such a response read, "Well, because of the `[' on the left side, we know $a$ is included in the interval. Because of the ')' on the right side, $a$ must also be excluded from the interval. But this doesn't make any sense. I cannot answer the question --- it seems the notation doesn't define an interval."

(Only a few students were this perceptive --- more often they decided 1) the paradox of $a$ being both included and excluded at the same time is not a paradox --- it simply means $a$ is not included in the interval, or 2) that the '[' somehow "trumps" the ')' so that $a$ is included in the interval).

I was a little disappointed, because I had asked them to read a chunk of the book before coming to class that first day and to pay special attention to definitions, especially ones they are already familiar with. Only a handful of students used the definition in our book:

$$ [a,a) = \{ \, x \in \mathbb{R} \mid a \leq x < a \, \} $$

from which they quickly, easily, and correctly decided that $a$ is certainly not in the interval $[a,a)$.

My main theme this year is getting these students to quickly recognize the importance of knowing and using definitions. I didn't even realize when I assigned the above problem that it would be such a good example of this --- being day one, I just wanted to see if they understood their reading about set notation in the book.

We will get to the epsilon-delta definition of limit soon, where precise definitions allow access to a host of pathological examples where student intuition will fail.

My question is: what are other examples of definitions besides the $[a,a)$ and $(a,a]$ notations and the definition of limit, where common student understanding of definitions can limit the questions that they are able to answer about them?

Ideally, these should be at the level of a typical beginning college math major. I don't want examples of definitions that can simply be moved to broader contexts, like switching from real-valued functions of real variables to functions between sets. Rather, within a single context, I want examples of two equivalent definitions that give the same meaning to common examples but where one is clearly weaker at deciding some pathological cases.

Edit: Here's another example: some students think that when working with functions, the input to every function is always "what's inside the parentheses". When asked to show $\cos(2\pi x)$ is periodic with period 1, these students immediately write $\cos(2\pi x + 1)$ and try to show it equals $\cos(2\pi x)$.

Edit: There are differing opinions about a typical beginning college math major. Let me change my question to allow atypical math majors. I don't mind examples that might not be covered before college, say from calculus or an area like combinatorics that might be accessible to strong math majors. I just want to avoid a definition at such a high level that a student would already need an appreciation of definitions to progress far enough to encounter said definition.

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    $\begingroup$ The input of a function is "what is left of the $\mapsto$ symbol"; you should have asked about $x\mapsto\cos(2\pi x)$ (or about the constant function $y\mapsto\cos(2\pi x)$ if that is what you meant ;-). $\endgroup$ – Marc van Leeuwen Sep 9 '12 at 13:13
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    $\begingroup$ Of course I meant the function $f \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \cos(2 \pi x)$. I think it is clear to my intended audience here what this type of problem says, and concision is a virtue. (Given the length of my post, a virtue that I lack. So perhaps you are right to bust my chops :-)) $\endgroup$ – Barry Smith Sep 9 '12 at 13:22
  • $\begingroup$ I think this should be community wiki'd $-$ there are likely to be lots of good responses. $\endgroup$ – Clive Newstead Sep 9 '12 at 14:19
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A personal pet peeve of mine:

  1. A set $C$ is countable if there is an injection $f:C\to\mathbb{N}$.

  2. A set $C$ is countable if there is a surjection $f:\mathbb{N}\to C$.

The first definition is the right one, the second one covers all countably infinite and nonempty finite sets. But it excludes the empty set, which is rightly covered by the first definition.

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A "prime" is a positive integer that is not the product of two smaller positive integers.

versus

A "prime" is a positive integer that has exactly two different divisors among the positive integers.

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  • $\begingroup$ Which one is inadequate? $\endgroup$ – Antonio Vargas Sep 9 '12 at 18:12
  • $\begingroup$ @AntonioVargas: The first one is "inadequate" in that it makes 1 a prime. $\endgroup$ – Henning Makholm Sep 9 '12 at 18:52
  • $\begingroup$ Would it be adequate if it said "A 'prime' is a positive integer different from 1 that is not a product of two smaller positive integers"? $\endgroup$ – Antonio Vargas Sep 9 '12 at 18:53
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    $\begingroup$ @AntonioVargas: Yes, and then it would be equivalent to the "exactly two divisors" definition. My point is that it is fairly easy to erroneously internalize the first definition as one's understanding of "prime", and students are likely to have learned about primes at a stage where exact explicit definitions were not emphasized. That could well lead them with a concept of prime that is cumbersome for number theory (in particular, prime factorizations are no longer unique up to reordering, and so the multiplicity of any prime in the factorization of a number is not well defined). $\endgroup$ – Henning Makholm Sep 9 '12 at 18:59
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    $\begingroup$ Actually the most common inadequate definition of primes I've heard (which also would include the $1$ as prime) is: "A prime number is a number which is divisible only by $1$ and by itself." $\endgroup$ – celtschk Sep 18 '12 at 16:33
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The tangent line to a curve $C$ at a point $P$ is:

The line passing through $P$ that intersects $C$ in just that one point

or

The line passing through $P$ such that $C$ stays on one side of the line

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  • $\begingroup$ neither one in general $\endgroup$ – Berci Oct 4 '12 at 15:41
  • $\begingroup$ K$\quad\quad\quad$ $\endgroup$ – Chris Taylor Oct 4 '12 at 16:04
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    $\begingroup$ That is the point. This are both common inadequate definitions. $\endgroup$ – Barry Smith Oct 4 '12 at 18:46
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A lot of students (and even some educators I know!) don't understand that a function might be neither even nor odd. Since the language is similar to a property of integers, they instinctively carry some other rules of thumb with it.

And while I'm thinking of it, a surprising number of people don't know whether 0 is even or odd. That's still hard for me to understand why this causes people difficulty.

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  • $\begingroup$ It's even more common, I think, to assume that a subset of $\mathbb R$ (or another topological space) must be either open or closed. $\endgroup$ – Henning Makholm Sep 18 '12 at 12:27
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    $\begingroup$ I think that's partially because of bad naming. I have no idea who came up with the term "even" and "odd" function (I guess the intuition is that even powers of $x$ are even functions, and odd powers of $x$ are odd functions). I personally prefer the names "symmetric" and "antisymmetric" function, because that better (although still not precisely) describes what it means (invariant resp. changing sign under the operation $x\mapsto-x$). Those names also directly imply that they are both special, so that the majority of functions is neither. $\endgroup$ – celtschk Sep 18 '12 at 12:29
  • $\begingroup$ @celtschk Someone I once worked with, who we hired to help write part of a new edition of a high school math textbook, actually stated that even functions are polynomial functions of even degree, and odd functions are polynomial functions of odd degree! Clearly, it's hard getting good help sometimes. $\endgroup$ – Richard Sullivan Sep 18 '12 at 16:17
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    $\begingroup$ Determining whether 0 is even or odd has a bit of an analogue in functions, where the constant function 0 is both even and odd, which is of course immediate from the definitions but can still confuse occasionally. $\endgroup$ – Logan M Sep 18 '12 at 16:28
  • $\begingroup$ Some students think a function continuous on a closed interval that has a maximum at a point cannot also have a minimum at that point. Those same students think that a constant function has neither a maximum value nor a minimum value. $\endgroup$ – Barry Smith Oct 4 '12 at 19:03
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Many definitions of limit I read around use the symbolism

$$ \lim_{x\to x_0}f(x)=l $$

before proving or even mentioning the uniqueness of the limit. If it was correct, the Theorem of Uniqueness of Limit would reduce to a simple application of the transitive property of equality

$$ \lim_{x\to x_0}f(x)=l_1\text{ and }\lim_{x\to x_0}f(x)=l_2 \implies l_1=l_2 $$

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It's a little late, but anything degenerate would be a good example.

  • How many ways to arrange 0 objects?
  • How many elements in $\{ 3, 3\}$?
  • What is a solution for $x$ in the equation $1+2=3$?
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Wow, I thought there'd be more answers. Is it that most students who know a definition at all know something equivalent to the precise definition? Or is it that mathematicians are so used to knowing and using only the most precise definitions that it is difficult to think of bad ones?

An integer $d$ is a divisor of an integer $a$ if there is an integer $n$ with $dn=a$.

According to this definition, $0$ is a divisor of $0$. This is easy to fix if we replace "an integer $n$" with "a unique integer $n$". I've never explored division by $0$, so I don't what horrible consequences, if any, come from using the above flawed definition.

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  • $\begingroup$ I think the most likely explanation is that "at the level of a typical beginning college math major" feels like a rather tight restriction. (For example, my first instinct was to write something about thinking L'Hospital's rule is the definition of the limit of fractions when the numerator and denominator go towards 0 -- but that doesn't qualify level-wise). $\endgroup$ – Henning Makholm Sep 18 '12 at 12:36
  • $\begingroup$ Ah, you're probably right. I don't mind pulling things out of calculus, like you did, since many of my students have had calculus before college. I'll edit my post. $\endgroup$ – Barry Smith Sep 18 '12 at 16:01
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    $\begingroup$ $0$ is a divisor of $0$. There's nothing wrong with this definition at all. $\endgroup$ – Chris Eagle Oct 4 '12 at 15:12
  • $\begingroup$ That fact makes this a good example: is the student using the notion of divisor, or have they replaced it with the idea "if I divide them, I get an integer?" $\endgroup$ – Hurkyl Oct 4 '12 at 15:58
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    $\begingroup$ There a plenty of reputable authors who disagree; 0 is not a divisor of 0 by convention. $\endgroup$ – Barry Smith Oct 4 '12 at 19:01

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