Prove/disprove given statement(s) is/are true/false

Question

Let $\Theta(x, y, z)$ be the statement $“x+y=z”$ and let there be two quantifications given as:

$(i). \forall x \forall y \exists z, \Theta(x, y, z)$

$(ii). \exists Z \forall x \forall y, \Theta(x, y, z)$

where $x, y, z$ are real numbers. Then which one of the following is/are TRUE ?

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My attempt:

Well, known statement $(i)$ is true, and $(ii)$ is false.(I agreed).

And somewhere, it explained as : $x+y=z$ for all $x$ for all $y$ there exist some $z$ which will satisfy this equation as e.g $x=4351$ $y=1111$ then some $z =5462$ is there and so on.

for some $z$ say $z=100$ there do not exist all $x$ and all $y$ (there exist only some $x, y$ ) which satisfies this equation hence $(ii)$ is false.

OK, my question is: but how we prove in mathematical way?

Can you explain it, please?

Answer 1

$(i)$ is true due to real number is closed under addition which is an axiom.

$(ii)$ is false.

Suppose such $z$ exists, then $z=0+0$ and $z=1+0$, hence $1=0$ which is a contradiction.