What is the negation of 'given' in a quantified statement?

Question

Suppose I have the following statement: Given any real numbers a and b, if a and b are rational, then $a/b$ is rational. Is the word given the same as $\forall$? If so, this would be a universal conditional statement, whose negation is "$\exists x$ such that $P(x)$ and ~$Q(x)$". It intuitively makes sense to me for given to sound like $\forall$, but I do not want to make any assumptions.

Answer 1

Indeed, "Given any real numbers $a$ and $b,$ if $a$ and $b$ are rational, then $\,a/b\,$ is rational," means $$\forall a \in \mathbb R, \forall b\in \mathbb R\, \Big(\big(a\in \mathbb Q \land b \in \mathbb Q\big)\to \frac ab \in \mathbb Q\Big)$$

The key word in the natural language statement is not "given", but rather "any."
So you are correct that the phrase "given any" means "for all" in this case.

And indeed, it's negation would be: $$\exists a, b \in \mathbb R \Big((a, b \in \mathbb Q)\land \left(\frac ab \not \in \mathbb Q\right)\Big)$$