Help with interpreting specific logical statements discrete math For a problem on a discrete math assignment, I am asked to find which of the following statements is true, but I am unsure if I'm interpreting it correctly as I don't know how to interpret every symbol yet. Here are the statements:
$$\exists ! x\in Z, \forall y\in Z, xy=x.$$
$$\exists !x\in Z, \forall y\in Z, xy=y$$
Should I interpret this as "there exists exactly one x for all y  such that ..." or should I interpret this as  "there exists exactly one x for each y  such that ..."
 A: I would consider the English interpretation of the above to be

*

*"There exists exactly one $x$ in $Z$, such that for any $y$ from $Z$, $x y = x$"

*"There exists exactly one $x$ in $Z$, such that for any $y$ from $Z$, $x y = y$"

It might be enlightening to consider what is required to actually prove statements like $\exists$ and $\forall$.
If you have to prove a statement like $\forall x \in Z. P(x)$, it means to show that $P(x)$ holds from an arbitrary $x$ taken from $Z$. For existentials, proving the statement $\exists x. P(x)$ requires you to come up with an $x$ from $Z$.
A statement of unique existence such as $\exists! x \in Z. P(x)$ requires you to both come up with an $x \in Z$ that satisfies $P$, but to also show that there is no other $x'$ that satisfies $P$ (or equivalently that any such element is, in fact $x$).
So, for example, the proof of first statement is going to look like:
Let $x$ be (some value from $Z$),

*

*Take any $y$ from $Z$, then $xy = x$ because of (reason).


*Take any $x'$ and any $y$ from $Z$, moreover, assume $x' y = x'$,
then $x' = x$ because of (reason).
