Common inadequate definitions
Solution 1:
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.
Solution 2:
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.
Solution 3:
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
Solution 4:
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.