Confusion with logic (Continued from "Is $1/(x-2)$ not differentiable at 2"?) I opened this question because I am still very confused by the answers and the comments from the following post: differentiability check.
In the problem, the OP asks the following simple question:
"What is the number of points at which $f(x) =\frac{1}{(x-2)}$ not differentiable?"
The OP himself and a number of people said that the correct answer is none, pointing out that differentiability of $f$ at $x=2$ is not well-defined, since $2$ doesn't belong to the domain of $f$.
I think the question here is ambiguous and it depends on the precise way "$f$ is differentiable at $x$" is defined. For example, if the definition is "$x$ is differentiable if and only if ($x$ is in the domain of $f$ and $f'(x)$ exists)," then $f$ is not differentiable at $2$. But then $f$ is also not differentiable at a 2x2 matrix, or any complex number (assuming that we defined the domain of $f$ to be $\mathbb{R}\setminus 2$).
However, if the question was
"What is the number of points in $\mathbb{R}$ at which $f(x) = \frac{1}{(x-2)}$ not differentiable?,"
then I believe that the answer is unequivocally 1. @GitGub said that merely adding the words "in $\mathbb{R}$" isn't sufficient to make a difference, but I don't really understand his argument. You can see his arguments here. 
Here's my reasoning:
Let $E = \{ x\in dom(f): \text{$f'(x)$ exists.} \}$. 
$2\in \mathbb{R}$, but $2\not\in E$. Therefore, $2\in \mathbb{R} \setminus E =\{2\}$.
The question is asking for the number of elements in $\mathbb{R} \setminus E$, which is 1.
I feel like I am missing some very basic understanding of logic. Can someone help me clarify this?

Edit: It doesn't feel like anyone is really addressing my reasoning. From the way I see it, the definition of "the set where $f$ is differentiable" is not the issue. Note the way I defined :
$E = \{ x\in dom(f): \text{$f'(x)$ exists.} \}$.
I have defined the set $E$, the set where $f$ is differentiable to be the points in the domain of $f$, as both @Vladimir, @Github, and @Hurkly is saying that some/many people would.
What I understood as the original issue was that just asking what $E^c$ (complement) is not well-defined, because there is no universe for the set $E$. When the universe is not specified, the most sensible interpretation would be to look at $dom(f) - E$.
The issue I am having is, if the question explicitly mentions $
\mathbb{R}$, "How many points in $\mathbb{R}$ is $f$ not differentiable," i.e. when a universe is given, why isn't this question logically equivalent to "How many points in $\mathbb{R} \setminus E$?"
Vladimir asked: "What if $g(x) =x$ was defined with $dom(g) = R\setminus\{2\}$."?
It doesn't change anything. $E = \{ dom(g): g'(x)\text{ exists}\}$ would be "$R\setminus\{2\}$", and $R\setminus E = \{2\}$.

EDIT2: I feel like reference to calculus is detracting the entire discussion.
Here is what I believe to be the essence of my question.
I have sets $A$ and $B$, with $A\subset B$, a proper subset.
Let $P$ be a predicate such that $P_A(x)$ is the sentence: "For $x\in A$, $x$ has property $P'$."
Let $E = \{ x\in A: P_A(x)\}$.
Then $E$ is a subset of $A$ and is also a subset of $B$. 
Take a point $y \in B$. 
I feel that @Hurkyl and @Github would say, for $y\not\in A$, $P_A(y)$ is not a well-defined statement. 
But then, why isn't asking "Is $P_A(y)$ true" equivalent to asking "Is $y \in E$"?
 A: What you are discovering is one of the little lies that crop up for the sake of simplification. We often talk about functions -- mathematical objects that associate to each point of its source a unique point in its target -- and define things like limits for functions.
But in practice, we very frequently are not dealing with functions: we are dealing with partial functions -- mathematical objects that associate to some points of its source a unique point, and to other points of its source no points at all.
(partial functions which are defined at every point of its source -- i.e. ones that are functions in the ordinary sense -- are called "total functions")
Your function $1/(x-2)$ is one of these: it is merely a partial function on $\mathbb{R}$, because it is not defined at $x=2$.
Now, there are a number of ways to deal with this issue. We could modify the definition of limit to work with partial functions. We could leave definitions unchanged but restrict our attention to the domain of the partial function (i.e. the set of points of the source where it is defined). We could even consider alternative objects, like functions modulo negligible functions.
Unfortunately, as far as I know, these things are rarely done explicitly in introductory courses -- and I'm not even sure what is done implicitly is even done in a consistent fashion.
Really, for a question like the one your topic is about, the answer requires guessing what the asker really means to ask, and there isn't a unique answer. I would guess it's most common to mean to ask the number of points for which $f$ is not defined plus the number of points for which $f$ is defined but not differentiable.
As an aside, in my opinion,the line of reasoning that involves "$f(x)$ is not defined at $2$ so it doesn't make sense to say if it is differentiable or if it is not differentiable there" really should be pushed further to say "it doesn't make sense to ask how many points of $\mathbb{R}$ that $f(x)$ is not differentiable at".
A: The ambiguity in the question "What is the number of points at which $f(x)=1/(x−2)$ is not differentiable?" resides not only in how you define "the domain of $f$" and "$f$ is differentiable at $x$" but also in the word "points". Once you define all three, everything becomes clear. If you define point to be an element of $R$, dom$f=R\setminus\{2\}$, and differentiable at $x\equiv (x\in\operatorname{dom}(f)$ and $f'(x)$ exists), then the answer is $1$. If you define points as elements of the set including real numbers, square matrices, and planets of the solar system, then the answer will be different;) 
In short: Much ado about nothing.
As an afterthought: let $g(x)=x$ with dom$(g)=R\setminus\{2\}$. What is the number of points $x\in R$ at which $g$ is not differentiable? Clearly, the answer (with the above definition of differentiability) is $1$. Much of the confusion in the previous discussion in this and the other topics stems from the fact that one (erroneously) asks about points $x$ an which $f(x)$ is differentiable. So apparently if $f(x)$ is undefined at that particular $x$, then the question makes no sense. But the right question is about points $x$ at which $f$ (the function), not $f(x)$ (a number, or an undefined entity), is differentiable.
