Should I use an existential quantifier here?

Question

In my question Proof Verification: Properties of Divisors, I had originally said something along the lines of the following:

Since $b \nmid (a+c)$ we have, by definition, $a+c \neq bq_2 \text{ for } q_2 \in \mathbb{Z}$.

But was told that I might have meant the following:

Since $b \nmid (a+c)$ we have, by definition, $ \nexists q_2\in \mathbb{Z} \text{ such that } a+c = bq_2$.

Is one of the previous more valid than the other and if so, why is it more appropriate to use? (i.e. Is there a difference between these two statements?)

Answer 1

Your first claim was not correct.

To correct it, you should indicate that the statement is universally true; like so:

Since $b \nmid (a+c)$ we have, by definition, $a+c \neq bq_2 \text{ for }\textbf{any } q_2 \in \mathbb{Z}$.

$$b\nmid(a+c)\iff \forall q_2\in\Bbb Z~.(a+c)\neq bq_2$$

Which is now equivalent to the second claim.

Since $b \nmid (a+c)$ we have, by definition, $ \nexists q_2\in \mathbb{Z} \text{ such that } a+c = bq_2$.

$$b\nmid(a+c)\iff \lnot\exists q_2\in\Bbb Z~.(a+c) = bq_2$$