Confused by this question about proving the equivalence (identicalness) of two equivalence relations

Question

I'm (self-studying, so I can't ask the author/professor, and) trying to figure out 1.8.4, which relies on 1.8.3. What's got me confused is:

1. Both these problems are referring to the same set X, right? And
2. $\approx$ is referring to a partition where, if $x \in X_\iota$, $[x]_{\approx} := \{ y \in X : y \in X_\iota\}$, right?

If so, is this how one would go about the proof?

$\Rightarrow$ Suppose $x \sim y$. Then $x, y \in [y]_{\sim}$ by the definition of an equivalence class. This is a subset of X and a member of a partition class of X. If we denote this subset $X_\iota$, , then we can see that $x \approx y$.
$\Leftarrow$ Suppose $x \approx y$. This means that $\exists \iota \in I : x \in X_\iota, y \in X_\iota$, which is the definition in 1.8.3 of $x \sim y$. This set $X_\iota = [x]_\sim = [y]_{\sim} $, so $x \sim y$.

Answer 1

Both of these problems are talking about an arbitrary set $X$. Technically, $\approx$ refers to an equivalence relation not a partition.

Your proof is correct.

Another way to represent this proof is to write $$ x \sim y \ \Leftrightarrow \ \exists \iota\in I: (x\in X_{\iota} \textrm{ and } y\in X_{\iota}) \ \Leftrightarrow x \approx y. $$

Answer 2

Exercise 1.8.4 is not very clear. It should say clearly that $\approx$ is the induced equivalence relation by the partition $X|_\sim$ as like in Exercise 1.8.3. With this you can show the conclusion, and this implies that both two $X$'s are the same one since the partitions are the same one. Also your proof is right.

By the way, it's more readable to write the conclusion in Exercise 1.8.3 and Exercise 1.8.4 in the following way.

Exercise 1.8.3. Let $X$ be a set. Show that, if $\mathcal{P}$ is a partition of $X$ then the relation

$$x\sim_\mathcal{P}y\stackrel{\mathrm{def}}{\Leftrightarrow} \text{there is some }P\in\mathcal{P}\text{ such that }x,y\in P$$

on $X$ is an equivalence relation.

Exercise 1.8.4. Let $X$ be a set. Show that, if $\sim$ is an equivalence relation on $X$ and $\mathcal{P}=X|_\sim$ then $\sim_\mathcal{P}\,=\,\sim$.

And in fact we can further have the following exercise.

Exercise 1.8.5. Let $X$ be a set. Show that, if $\mathcal{P}$ is a partition of $X$ and $\sim\,=\,\sim_\mathcal{P}$ then $X|_\sim=\mathcal{P}$.

Exercise 1.8.4 shows that any partition can be determined by only one equivalence relation, and Exercise 1.8.5 shows that any equivalence relation can produce only one partition.