Difference between mentioning existential quantifier and not

Question

Let $n \in N$, Is there a difference between:

1) let us assume as true $\exists k \in Z / n= 9 k$ and

2) let us assume as true $ n = 9k / k \in Z$?

Answer 1

You appear to be using a nonstandard notation with this "/" symbol. For instance, it is not listed as a logical symbol in Wikipedia, which only lists it as a division or quotient operator, and every other source I could find with a simple Google search revealed the same usage. Anecdotally, this is consistent with my experience as a student of math: the statements that you've written in the question (and in the Nov. 19 comment) parse as invalid.

However, I am going to exercise some judgement here. It seems reasonable to assume that this symbol roughly means "such that":

(1) Let us assume as true $\exists k\in\Bbb Z$ such that $n=9k$.

(2) Let us assume as true $n=9k$ such that $k\in\Bbb Z$.

The second statement still reads a little awkward to me; if I came across it in a research article I would assume it meant

(2*) Let us assume as true $n=9k$ for $k\in\Bbb Z$.

In this case, (1) and (2*) read to me as the same statement. However, both of these are different from another reasonable interpretation of (2):

(2^) Let us assume as true $n=9k$ for all $k\in\Bbb Z$.

In this case, as Mauro mentioned in the comments, (2^) is different from (1), because, for instance: if $n=18$ then $n=9k$ only for $k=2$, but there are other numbers $k\in\Bbb Z$ besides $2$, for which $n=9k$ is not true.