How many equivalence relations over $\mathcal P(\mathbb N)$ satisfy: $[\{8\}]_S=\{A\in \mathcal P(\mathbb N)|A\neq \{1\}\wedge A\neq \{2\}\}$

Question

How many equivalence relations $S$ over $\mathcal P(\mathbb N)$ satisfy:

$$[\{8\}]_S=\{A\in \mathcal P(\mathbb N)\mid A\neq \{1\}\wedge A\neq \{2\}\}$$

Just to make sure I understand, the question asks to find the cardinality of a quotient set $\mathcal P(\mathbb N)/S$?

Then I can build a bijection: $f(X)=\begin{cases}\{n+3\} &,X=\{n\}\\ X &,else \end{cases}$ which satisfy the above condition so the cardinality is $\aleph$.

Answer 1

There is a natural bijection between equivalence relations and the partitions generated by the classes, so we can equivalently ask how many different partitions of ${\cal P}(\Bbb N)$ contain $\{A\in {\cal P}(\Bbb N)\mid A\neq \{1\}\land A\neq \{2\}\}$ as a member. (The part about $\{8\}$ is not relevant because clearly $\{8\}$ is in the above set, it is just showing one representative of this equivalence class.)

So we are left to partition the remainder of ${\cal P}(\Bbb N)$ after taking away everything in this set. The remainder is $\{A\in {\cal P}(\Bbb N)\mid A=\{1\}\lor A=\{2\}\}=\{\{1\},\{2\}\}$, and there are only two ways to partition this, as one set and as two sets, yielding two partitions:

$$P_1=\{\{\{1\}\},\{\{2\}\},\{A\in {\cal P}(\Bbb N)\mid A\neq \{1\}\land A\neq \{2\}\}\}$$ $$P_2=\{\{\{1\},\{2\}\},\{A\in {\cal P}(\Bbb N)\mid A\neq \{1\}\land A\neq \{2\}\}\}$$

which correspond to the equivalence relations

$$x\,S_1\,y\iff x=y=\{1\}\lor x=y=\{2\}\lor x,y\in\{A\in {\cal P}(\Bbb N)\mid A\neq \{1\}\land A\neq \{2\}\}$$ $$x\,S_2\,y\iff x,y\in\{\{1\},\{2\}\}\lor x,y\in\{A\in {\cal P}(\Bbb N)\mid A\neq \{1\}\land A\neq \{2\}\}.$$

Answer 2

There are only two of them, precisely those for which equivalence classes are given by $$ \{ \mathcal{P} (\mathbb{N})\setminus \{ \{ 1\}, \{ 2\}\}, \{\{ 1\}\}, \{\{ 2\}\}\}$$ and $$ \{ \mathcal{P} (\mathbb{N})\setminus \{ \{ 1\}, \{ 2\}\}, \{\{ 1\}, \{ 2\}\}\}$$