Why is $\exists x \forall y \exists z \left(\left( y = x + z \right) \lor (z \leq x) \right)$ false?

Question

I am trying to wrap my head around the proposition $$\exists x \forall y \exists z \left(\left( y = x + z \right) \lor (z \leq x) \right),$$

where $x, y, z \in \Bbb N^+$.

The proposition is false, but I don't understand why. It is stated as

There is a value $x$ which for every value $y$, there is a value $z$ such that ( = + ) ∨ ( ≤ ),

so I started off with $(x,y,z) = (1,2,1)$, which is true.

Next, I tried $(x,y,z) = (1,3,2)$, which makes

(3 = 1 + 2) ∨ (2 ≤ 1)

T ∨ F = T.

I kept going, but I keep getting True.

Answer 1

It's true since taking $z=x$ the right-hand side (wrt the $\lor$) of the statement $y=x+z \ \lor \ z \leq x$ is always true. It is surely a typo.

Answer 2

$y$ can be any positive integer. Also, it says that there exists an $x$, so I can pick whatever $x$ I want. Pick $x=1$. Then $y=x+z$ is satisfied for $z=y-x$. However, $z$ must be a positive integer which is not the case only when $x \geq z $. The statement is clearly true.