So I am working on this math problem which states that for every x (domain: all integers) there is a y (domain: all integers) for which x=1/y is true.
∀x∃y (x=1/y)
If I understand this correctly, it means that there is some y for every x in the set of integers which makes x=1/y always true. This is false, but I don't understand how to show a counterexample for this one. I could try explaining how this is not false but aside from that, I am not sure what to do.
On
To stress how your understanding is wrong, you're doing the qualifiers in the word order. $\forall x \exists y P(x,y)$ is almost never logical equivalent to $\exists y \forall x P(x,y)$. This says that for every integer there is some integer that is the inverse, not the reverse as you've said. Both are false, but they're very different sentiments.
This is not good understanding. The formula says that any $x\in\Bbb Z$ is the inverse of some $y\in\Bbb Z$. This is evidently false. The counterexample is $x=2$. There is no $y\in\Bbb Z$ s.t. $2=\dfrac{1}{y}$ (if should be $y=\dfrac{1}{2}\not\in\Bbb Z$). So, the negation is true: there exists $x\in\Bbb Z$ ($x=2$) s.t. for all $y\in\Bbb Z$ we have $x\ne\dfrac{1}{y}$.
My answer is good, if the domain for $y$ is the set of all integers. Nevertheless, if the domain for $y$ is any subset of raeals, it is enough to take $x=0$ for the counterexample.