Negating the definition of a limit point Below is a definition of a limit point:

$E$ is a subset of a metric space $X$. $p \in X$ is a limit point of $E$ exactly when every ball around $p$ has an element $q \in E$ such that $q \neq p$.

Now, I have read online that the negation of this is:

$p \in X$ is not a limit point of $E$ exactly when there is some ball around $p$ whose every element $q$ such that $q \neq p$ does not lie in $E$.


I do not really understand why we do not negate the statement $q \neq p$, according to the laws of logic? Why is that statement left invariant during the negation? 
 A: Let us write the definition of a limit point using a first order logic formula:
$$\forall B \text{ ball around } p, \exists q \in B, (q\neq p\land q\in E).$$
Now we negate this:
$$\exists B\text{ ball around p}, \forall q \in B, (q=p\lor q\not\in E).$$
This is not quite statement you were given. But note that $(q=p\lor q\not\in E)$ is equivalent to $(q\neq p\Rightarrow q\not\in E)$. Thus, we finally obtain
$$\exists B\text{ ball around p},\forall q \in B, (q\neq p\Rightarrow q\not\in E).$$
Translating this to English gives us "There is some ball around $p$ whose all points $q$ such that $q\neq p$ are not in $E$", which is what you were given.
A: I love the answer that zarathustra gave, but I found it rather symbol-heavy; it is rigorous, but not perfectly clarifying to some (myself included). I see no harm in adding a second answer, especially since it is for the purpose of clarity, so I will do so.
Limit Point

$E$ is a subset of a metric space $X$. $p\in X$ is a limit point of $E$ exactly when every ball around $p$ has an element $q\in E$ such that $q\not=p$.

Non-limit Point

$p\in X$ is not a limit point of $E$ exactly when there is some ball around $p$ whose every element $q$ such that $q\not=p$ does not lie in $E$.

From these, the keywords are as follows, for Limit and Non-limit points, respectively:

...exactly when every ball around $p$... [emph. added]
  
  ... when there is some ball around $p$... [emph. added]

This states that a limit point must have each and every, whilst the non-limit point must have some; translated, it must have at least one.
The statement $q\not=p$ is left the same in both definitions because it necessitates a certain attributed of $X$.
