TITLE: What are the most misleading alternate definitions in taught mathematics?
QUESTION [143 upvotes]: I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent definitions where one is more natural or intuitive? (I'm meaning so greatly more intuitive so as to not be subjective.)
Alternatively, what common examples are there in standard lecture courses where a particular symbolic definition obscures the concept being conveyed.

REPLY [7 votes]: Ok I'm joining very late but let me tell this.
I think the following definition of relation, that is often given, is misleading:

"Wrong" definition: A relation $R$ between the set $A$ and the set $B$ is an arbitrary subset of the cartesian product, $R\subseteq A\times B$.

In fact, I think it is a "wrong" definition:

it obfuscates the fact that the datum of source set and target set is important and itself part of the definition. For example, if you defined a function as a relation (in the sense of the definition above) satisfying the functional property ($\forall x\in A\exists ! y\in B: (x,y)\in R$) then the notion of codomain will not be well defined (or not explicitly defined), and one could be led to think that $x\mapsto x^2$ as a function $\mathbb R \to [0,+\infty)$ is literally equal to $x\mapsto x^2$ as a function $\mathbb R\to \mathbb R$.
It allows you to define, given $A$ and $B$ (in that order), the set $\mathsf{Rel}(A,B)$ of relations between $A$ and $B$, but not (immediately) the class of all relations (or the set of all relations within a given universe $U$).
It makes less clear that there should be a category $\mathsf{Rel}$, of sets with relations as morphisms, of which the $\mathsf{Rel}(A,B)$ are the hom-sets.

The right definition should of course be:

"Right" definition: A relation is a triple $(A,B,R)$ where $A$, $B$ are sets and $R\subseteq A\times B$.

Now the notion of codomain is well defined (or explicitly defined). And also the category $\mathsf{Rel}$ is well defined.