Translation question from english to symbolic sentence? Universe the real numbers.


*

*between any integer and any larger integer there is a real number.


$\forall x \forall y$ $(x \in Z \wedge y \in Z \wedge y>x \rightarrow \exists k (x<k<y)$


*

*Every nonzero real number is positive or negative.


$\forall x(x\not=0 \rightarrow x>0 \lor x<0$
3.
All people hate some taxes.
$\forall x \exists y$( if x is a person and y is a tax then x hates y)

You need to fix (add) parentheses, and on number $(1)$, you need to say that $k \in \mathbb R$:
$$\forall x\forall y\Big((x \in \mathbb Z \wedge y \in \mathbb Z \wedge y>x )\rightarrow \exists k (k \in \mathbb R \land x< k <y)\Big)$$
Same thing for $(2)$. Here we need to specify $x \in \mathbb R$:
$$\forall x\Big((x\in \mathbb R \land x\not=0) \rightarrow (x>0 \lor x<0)\Big)
$$
For $(3)$, $$\forall x\Big(\text{Person}(x)\rightarrow \exists y(\text{Tax}(y) \land \operatorname{Hates}(x, y))\Big)$$
