How do you negate: $ \forall x\in \mathbb{R},\exists n,m\in \mathbb{Z} \hspace{3 mm}s.t.\hspace{1 mm}m
Here's what I got, but it's obviously flawed. $ \exists x \in \mathbb{R} \hspace {2 mm} s.t. \forall n,m \in \mathbb{Z} \hspace{1mm}, \neg (m < x \hspace{1.5mm}and\hspace{1.5mm} x
I got: $x\leq m \hspace{4mm} OR \hspace{4mm} n \leq x$
which is obviously not what the author of the book intended.
Another way for $\neg (m < x \hspace{1.5mm}and\hspace{1.5mm} x
$\forall m,n \in \mathbb{Z}, (m \leq x\hspace{2mm} and \hspace{2mm} n \leq x) or (m \geq x \hspace{2mm} and \hspace{2mm}n \geq x)$
but I'm not so sure about the alternative either. (it's somewhat intuitively right to me, but it might not)
Could you point out where I'm wrong, and how to fix that?
Thanks :D