Cardinality of equivalence relations in $\mathbb{N}$

Question

I came across this long proof on this site:

Cardinality of relations set

But I would like to know whether my direction can work.

Say we want to find the cardinality of all equivalence relations in $\mathbb{N}$. Since it is a subset of all relations in $\mathbb{N}$, I conclude it has a cardinality smaller or equal to $\aleph$. Now, define an injective function from $P(\mathbb{N})$ to the set of equivalence relations by matching each subset of $\mathbb{N}$ with the identity relation (which is an equivalence relation in $\mathbb{N}$.

Therefore the cardinality of all equivalence relations in $\mathbb{N}$ is greater or equal to $\aleph$ and using CSB we get the desired result.

Seems legit?

Answer 1

Your approach is sound. I think you can choose a better injection from $P(\mathbb{N})$ into equivalence relations. Let us say $0 \in \Bbb N$ and we will inject $P(\Bbb N \setminus\{0\})$ into the equivalence relations on $\Bbb N$. I would suggest you take a subset of $\Bbb N \setminus\{0\}$ to the equivalence relation that groups all members of the subset and $0$ into an equivalence class and leaves all the rest of $\Bbb N$ under the identity.

If we don't do the trick with $0$, all singleton subsets will be mapped to the identity relation and you don't have an injection.