Truth of statements about numbers

Question

Given the following statements:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

16. $\forall\,m\in\Bbb Z\quad\exists\,q\in\Bbb Q:m\lt q\lt m+1$

list which are true.

Only $2,6$ and $8$ are false, correct?

Answer 1

The complete list of true statements is:

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

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

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

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

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

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

14.) $\forall\,x,y\in\Bbb Z\quad\exists\,z\in\Bbb Z:x+z=y$

16) $\forall\,m\in\Bbb Z\quad\exists\,q\in\Bbb Q:m\lt q\lt m+1$

For each the others (that are not true), you need only find a single counterexample in which the statement is false: $\color{red}{1, 2,4,6,8,10, 12, 15}$

Note that $1)$ is false whenever $x = y$. It would be true, however, if we have $$\forall x, y \in \mathbb Q((x\neq y) \rightarrow \exists z \in \mathbb Q (x\lt z \lt y)\lor (x\gt z\gt y))$$

$(12)$ is false, if we take $a = 0, b= 2$, e.g., but is true if we state 12) $\forall\,a,b\in\Bbb Q((a\neq 0)\rightarrow\exists\,x\in\Bbb Q(ax=b))$

Answer 2

Number 1 is false for x=3 and y=4 there doesn't exist a z.
Number 4 as well for x=0.
Number 10 for x=3 and y=2 there doesn't exist a z.
Number 15 take a=2 and b=1 there doesn't exist x.

Answer 3

1. False. Let $x:=0$ and $y:=0$.
2. False. Let $x:=-1$.
3. True. Let $y:=\sqrt{x}$.
4. False. Let $x:=0$.
5. True. Let $z:=\frac{x+y}{2}$.
6. False. Let $x:=0$.
7. True. Let $y:=\sqrt{|x|}$.
8. False. Let $x:=0$.
9. True. Let $x:=0$.
10. False. Let $x:=1$ and $y:=0$.
11. True. Let $y:=x+\frac{1}{2}$.
12. False. Let $a:=0$ and $b:=1$.
13. True. Let $y:=x-1$.
14. True. Let $z:=y-x$.
15. False. Let $a:=0$ and $b:=1$.
16. True. Let $q:=m+\frac{1}{2}$.

Answer 4

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