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
(iii) true (not sure about this one)
(iv) true (not sure about this one)
(v) true (not sure about this one)
Basically, the book just says that the order of quantifiers matters and that one statement is true and the other is not; there's no real explanation of how to interpret it, though. Any input would be appreciated. Thanks!