Truth of statements about numbers Given the following statements:


*

*$\forall\, x,y \in \Bbb Q \quad \exists\, z \in \Bbb Q $ $\;$ $ : \left(x<z<y\right) \vee \left(x>z>y\right)$.

*$\forall \, x \in \Bbb R \quad \exists\,y\in \Bbb R : y^2= x$

*$\forall \, x \in \Bbb R^+ \quad \exists\,y\in \Bbb R : y^2= x$

*$\forall \, x \in \Bbb Z : | x | > 0$

*$\forall\, x,y \in \Bbb Q : \left(x<y \rightarrow \exists\, z \in \Bbb Q: x<z<y \right)$

*$\forall \, x \in \Bbb N \quad \exists\, y \in \Bbb N : x>y$

*$\forall \, x \in \Bbb R \quad \exists\, y \in \Bbb R : y^2=|x|$

*$\forall \, x \in \Bbb Z \quad \exists\, y \in \Bbb Z: x \lt y \lt x+1 $

*$\exists\,x \in \Bbb N \quad \forall\, y \in \Bbb N : x\le y$

*$\forall \, x,y \in \Bbb N \quad \exists \, z \in \Bbb N: x+z=y$

*$\forall\,x\in\Bbb R\quad\exists\,y\in\Bbb R:x\lt y \lt x+1$

*$\forall\,a,b\in\Bbb Q\quad\exists\,x\in\Bbb Q:ax=b$

*$\forall\,x\in\Bbb Z\quad\exists\,y\in\Bbb Z : x\gt y$

*$\forall\,x,y\in\Bbb Z\quad\exists\,z\in\Bbb Z:x+z=y$

*$\forall\,a,b\in\Bbb Z\quad\exists\,x\in\Bbb Z:ax=b$

*$\forall\,m\in\Bbb Z\quad\exists\,q\in\Bbb Q:m\lt q\lt m+1$
list which are true.
Only $2,6$ and $8$ are false, correct?

1 isn't true as I've pointed out in the comments, as $x=y$ gives a counterexample; 
2 of course isn't true; 
3 is true; 
4 is obviously wrong; 
5 is the correct version of 1, it's true; 
6 is obviously wrong; 
7 is true; 
8 is wrong; 
9 is true; 
10 is wrong; 
11 is right; 
12 is wrong, take $a=0, b \neq 0$; 
13 is true; 
14 is true; 
15 is wrong, there are the counterexamples of 12, and even more; 
16 is true; 
So the list of false statements is more than OP thought, it's actually 1, 2, 4, 6,8,10,12,15
