Are my quantified statements translated from English correct?

Question

There exists a real number between any two sequential natural numbers.

$$(\exists x \in \Bbb {R})( \forall a,b \in \Bbb {N}) a < b, b > x > a$$

If an integer is positive, it equals its absolute value

(∀xℤ+)[x=|x|]

Answer 1

Your statement is incorrect - it gives one $x$ for all $a, b \in\mathbb{N}$. You need exchange positions of quantifiers i.e. $$( \forall a,b \in \Bbb {N}, a < b)(\exists x \in \Bbb {R})( b > x > a)$$