# Order of Existential/Universal Quantifiers

I just wanted to check my answers for this because I'm still not that comfortable with it.

Which of the following statements are true?

(i) $(\forall x \in \mathbb R)$ $x+1>x$

(ii) $(\forall x \in \mathbb Z)$ $x^2>x$

(iii) $(\exists x \in \mathbb Z)(\forall y \in \mathbb Z)$ $x \le y$

(iv) $(\forall y \in \mathbb Z)(\exists x \in \mathbb Z)$ $x \le y$

(v) $(\forall \epsilon > 0)(\exists \delta >0)(\forall x \in \mathbb R)$ $[0<\lvert x-1 \rvert < \delta] \implies [\lvert x^2-1\rvert<\epsilon]$

I have:

(i) true

(ii) false

• Perhaps it would be better to interpret (iii) as "There exists $x \in \mathbb{Z}$ such that, for all $y \in \mathbb{Z}$, $x \leq y$." So, does there exist such an $x$? – Ken Feb 18 '15 at 21:53
• Yes! Now, we interpret (iv) as "For every integer $y$, there exists an integer $x$ such that $x \leq y$." What do you think here? – Ken Feb 18 '15 at 22:33