Turning a definition in English to one using quantifiers questions. Real Analysis

Question

so the book im studying defines adherent points in the following way.

Let $X$ be a subset of $R$, and let $x \in R$ We say that $x$ is an adherent point to $X$ iff for every $\epsilon > 0$ there exists a $y\in X$ which is $\epsilon-close$ to $x$ (i.e $|x-y|\leq\epsilon$)

so in terms of quantifiers which one does it go along with is it either:

$\forall\epsilon >0\exists y$ $( y\in X \rightarrow |x-y| \leq \epsilon$

or is it

$\forall\epsilon >0\exists y$ $( y\in X \land |x-y| \leq \epsilon)$

since i am trying to negate them in which case the first one becomes

$\exists\epsilon >0\forall y$ $( y\in X \land |x-y| > \epsilon)$

and the second becomes

$\exists\epsilon >0\forall y$ $( y\notin X \lor |x-y| > \epsilon)$

which seems strange please any thoughts? thank you so much!

Answer 1

These formalisations of the given definition are all correct (and logically equivalent): \begin{gather}\forall\epsilon \;\Big(\epsilon >0 → \exists y\; \big(y\in X ∧ |x-y| \leq \epsilon\big)\Big),\tag1\\ \forall\epsilon {>}0\;\exists y{\in} X \quad |x-y| \leq \epsilon,\tag2\\ \forall\epsilon \;\exists y \;\Big(\epsilon >0 → \big(y\in X ∧ |x-y| \leq \epsilon\big)\Big).\tag3\end{gather}

Statement $(2)$ is an abbreviation of statement $(1).$ Its negation is $$∃\epsilon {>}0\;∀y{\in} X \quad |x-y| > \epsilon.$$

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Addendum (from the comments)

After your edit, your second and fourth formalisations are indeed correct.

the blank in between y∈X and |x−y|>ϵ seems strange.

I widened the gap/spacing merely for readability. Do note that $$∃y{∈}X\;P(y)\tag A$$ is just a more succinct way of writing $$∃y\;\Big(y∈X\land P(y)\Big).\tag B$$

right so when we negate it we get $$\forall y (y \notin X \lor P(y)) \equiv \forall y \in X ~P(y);$$ is that correct?

I think you meant to write $$\forall y\;(y \notin X \lor ¬P(y))$$ and $$\forall y{\in}X\;¬P(y)$$ instead, which are indeed the correct (and logically equivalent) negations of $(B)$ and $(A).$