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Given the following statements:

  1. $\forall\, x,y \in \Bbb Q \quad \exists\, z \in \Bbb Q $ $\;$ $ : \left(xz>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

  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?

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    No, there are others that are false. For instance, 1 is false. Ex: let x=1, y=2. Then there is no z satisfying the conditions. – 2017-01-26
  • 1
    4. is false when x=0. – 2017-01-26

4 Answers 4

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

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) $\forall\,a,b\in\Bbb Q((a\neq 0)\rightarrow\exists\,x\in\Bbb Q(ax=b))$

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    I just realized i goofed up when I transcribed my list, statement $1.$ should have been about rational numbers such that $\forall\, x,y \in \Bbb Q \quad \exists\, z \in \Bbb Q $ $\;$ $ : \left(xz>y\right)$ wihch is why I said $1.$ was true, can I add it to your answer? – 2017-01-27
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    Yes, in the case of $$\forall x, y \in \mathbb Q(\exists z\in Q: (x\lt z\lt y)\lor (x\gt z\gt y))$$ is indeed true (for any $x, y, \in \mathbb Q).$ Just put $z = \frac {x+y}{2}$, – 2017-01-27
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    Statement 1. as phrased is still false. Take $x=y$ – 2017-01-27
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    Aha...Your right, @Max – 2017-01-27
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    So 1 and 12 should be added to your list – 2017-01-27
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    @amWhy: So statements involving quantifiers $\exists$ and $\forall$ are always proved or disproved by finding counterexamples? – 2017-01-27
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    Typically , quantified statements are *disproved* when a counter-example exists. In true cases, you can often justify why true, e.g. $(3)$: For every $x\in \mathbb R$, there exists $y = \sqrt x$, such that $y^2= x$.... – 2017-01-27
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    That should be (in my last comment) " For every x \in \mathbb R^+, there exists y (namely, $y = \sqrt x,) $ such that $y^2 = x$" – 2017-01-27
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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.

  • 2
    It might also be good to confirm or disconfirm the OP's findings. – 2017-01-26
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  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 $q:=m+\frac{1}{2}$.
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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