Negation of a statement

Question

What would be the negation of the following statement? "There exist vertices $u$ and $v$ of $G$ such that the edge $x$ is on every path joining $u$ and $v$."

Would it be, "there exist vertices $u$ and $v$ of $G$ such that the edge $x$ is on every path joining $u$ and $v$" or "There does not exist vertices $u$ and $v$ of $G$ such that the edge $x$ is on every path joining $u$ and $v$"?

Thanks for your help!

Answer 1

Your second suggestion is correct: the negation of

there exists something such that so-and-so

is

there does not exist something such that so-and-so.

However, this can often be phrased more perspicuously. Saying that

there are no vertices $u$ and $v$ of $G$ such that the edge $x$ is on every path joining $u$ to $v$

is equivalent to saying that

no matter what vertices $u$ and $v$ of $G$ you look at, there is at least one path joining them that does not contain the edge $x$.

(And if you’ve learned a little graph-theoretic terminology, you can work out that this is just saying that if you remove the edge $x$ from $G$, the graph that remains is connected.)