Michael Hong wrote:

> It's quite difficult talking like a mathematician LOL.

True, at first. But once you learn how, it's very relaxing, sort of like knitting. You just work through things in a systematic, calm and precise way until everything fits neatly into place.

> **Puzzle 28.** Show that \\(\sim_P = \sim\\).

That's not what Puzzle 28 says. I don't even know what this means.

Puzzle 28 tells you to take a partition \\(P\\) and show that \\(\sim_P\\) is an equivalence relation. And that's what you actually did!

You started with the definition of \\(\sim_P\\):

> **Definition.** \\(x\sim_Py\\) if and only if \\(x,y \in S\\) for some \\(S \in P\\)

And then you checked that \\(\sim_P\\) was an equivalence relation:

> So we have to check if \\(\sim_P\\) satisfies **Reflexivity**, **Transitivity**, and **Symmetry** [....]

You also wrote:

> **Puzzle 29** : Show that \\(P_{\sim P} = P\\).

That's not what Puzzle 29 says. I don't even know what this means.

Puzzle 29 tells you to take an equivalence relation and show that \\(P_\sim\\) is a partition. And that's what you actually did. You started with the definition of \\P_\sim\\), namely the definition of its parts, and worked from there.

After you've done the real Puzzles 28 and 29, it _is_ very nice to show that

$$ \sim_{P_\sim} = \sim $$

and

$$ P_{\sim_P} = P $$

because these give you the one-to-one correspondence between partitions and equivalence relations.

The first one says that taking an equivalence relation \\(\sim\\), turning it into the partition \\(P_\sim\\), and then turning that back into an equivalence relation \\(\sim_{P_\sim}\\), gets us back where we started. The second one says that taking a partition \\(P\\), turning it into the equivalence relation \\(\sim_P\\), and then turning that back into a partition \\(P_{\sim_P}\\), gets us back where we started. Taken together, these mean we have a one-to-one correspondence.

There are also other ways to show we have a one-to-one correspondence. I didn't check your proof that there's a one-to-one correspondence between partitions and equivalence relations - I got a bit lazy. Someone else, please take a look! Is it okay?