I am trying to express the following statements in formal language and negating each of them.
The following is my attempted solution.
The universe of each statements is the set of real numbers and both $\varepsilon$ and $\delta$ denote positive real numbers.
Let $a$ be a fixed element of $\mathbb{R}$. For every $\varepsilon$ there exist $\delta$ such that for every $x \in \mathbb{R}$, if $|x-a| < \delta$, then $|x^{2} - a^{2}| < \varepsilon$.
For every $\varepsilon$, there exists an integer $N$ such that $1/n$ < $\varepsilon$ for all $n \geq N$.
Each of the followings is the expression in formal language for each of the statements respectively:
$\forall \varepsilon,(\varepsilon \in \mathbb{R}^{+} \rightarrow \exists \delta ,(\delta \in \mathbb{R}^{+} \wedge \forall x,(|x-a|<\delta \rightarrow |x^{2} - a^{2}|<\epsilon))).$
$\forall \varepsilon,(\varepsilon \in \mathbb{R}^{+} \rightarrow \exists N,(N\in \mathbb{Z} \wedge \forall n,(n \geq N \rightarrow 1/n < \varepsilon))).$
The respective negation of the each of the statements are the following:
$\exists \varepsilon ,(\varepsilon \in \mathbb{R}^{+} \wedge \forall \delta ,(\delta \in \mathbb{R}^{+} \rightarrow \exists x,(|x-a| < \delta \wedge |x^2 - a^2| \geq \varepsilon))).$
$\exists \varepsilon ,(\varepsilon \in \mathbb{R}^{+} \wedge \forall N,(N \in \mathbb{Z} \rightarrow \exists n,(n \geq N \wedge 1/n \geq \varepsilon))).$
Each of the above negations of the original statements can be expressed as plain English respectively as follows:
Let a be a fixed element of $\mathbb{R}$. There is a positive real number $\varepsilon$ and for every $\delta$, if $\delta$ is a positive real number, then there exists $x$ such that $|x-a| < \delta$ and $|x^2 - a^2| \geq \varepsilon$.
There exists a positive real number $\varepsilon$ and for every $N$, if $N$ is an integer, then there exists an $n$ such that $n \geq N$ and $1/n \geq \varepsilon$.
My questions are: Are all my expressions correct , and why in all of the cases, when one negates each of the statements, one does not negate: the definition "Let $a$ be a fixed element of $\mathbb{R}."$, $\varepsilon \in \mathbb{R}^{+}$, $\delta \in \mathbb{R}^{+}$, and $N \in \mathbb{Z}$?
All statements are form:
Daepp, U. and Gorkin, P., 2011. Reading, writing, and proving. 2nd ed. New York: Springer, p.46.
I would call these formulations of the statements and their negations correct. The extra subdomain quantifiers you mention could be avoided but you’d get a longer statement and it wouldn’t really affect the negation, in that $\forall \varepsilon \in \Bbb R^+ \phi$would be negated as $\exists \varepsilon \in \Bbb R^+ \lnot \phi$ etc. So leave them in I’d say.
The $a$ is what’s formally called a parameter of the formula; it need not be negated.