Common inadequate definitions 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.
 A: A personal pet peeve of mine:


*

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

*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.
A: 
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.

A: 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

A: 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.
A: 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
$$
\lim_{x\to x_0}f(x)=l_1\text{ and }\lim_{x\to x_0}f(x)=l_2
 \implies l_1=l_2
$$
A: 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 $1+2=3$?

A: 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.
