How to write the negation of sensitive dependence for initial conditions

Question

So I'm given this statement for sensitive dependence on initial conditions:

$$\exists \epsilon > 0 \, \, \forall x \in X \, \, \forall \delta > 0 \, \, \exists y \in X \, \, \exists n \in \mathbb{N}: \\ d(x,y) < \delta \text{ but } d(f^n(x), f^n(y)) > \epsilon.$$

I'm reading this to be: There exists $\epsilon > 0$ such that for all $x \in X$ and $\delta >0$, we can find $y\in X$ and $n\in \mathbb{N}$ such that $d(x,y) < \delta$ but $d(f^n(x), f^n(y) > \epsilon$.

So would the negation of this statement be: For all $\epsilon > 0$, we can find $x\in X$ and $\delta > 0$ such that for any $y\in X$ and $n \in \mathbb{N}$, $d(x,y) < \delta$ but $d(f^n(x), f^n(y) < \epsilon$.

Is that right?

Answer 1

$P \text{ but } Q$ must be translated with $P \text{ and } Q$, i.e.

$P \land Q$.

The negation of $P \land Q$ is $\lnot P \lor \lnot Q$, or alternatively: $P \to \lnot Q$.

Thus, the negated condition will be:

if $d(x,y)<δ$, then $d(f^n(x),f^n(y)) \le ϵ$.