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?