Equivalence relation and quotient set

Question

I'm studying for a test and got stuck in one question regarding equivalence relations and quotient set. Here is the question:

Let $F=\mathbb{R}\to \mathbb{R}$ be the set of functions from $\mathbb{R}$ to $\mathbb{R}$.

• Give an example of equivalence relation S in F such that $|F/S|=2^{\aleph}$
• Give an example of equivalence relation S in F such that $|F/S|=\aleph$
• Give an example of equivalence relation S in F such that $|F/S|$ is not $2^{\aleph},\aleph \ or \ 1$

I have no clue how to begin. I guess that an explanation of the first one will suffice.

Answer 1

Hint:

• Take any partition $\mathcal{F}_{2^{2^{\aleph_0}}}$ of $F$ into $2^{2^{\aleph_0}}$ non-empty sets (e.g. $\{\{f\} \mid f \in F\}$).
• Take any partition $\mathcal{F}_{2^{\aleph_0}}$ of $F$ into $2^{\aleph_0}$ non-empty sets (e.g. split by $f(0)$, that is, take sets $A_x = \{ f \mid f(0) = x\}$).
• Take any partition $\mathcal{F}_{\aleph_0}$ of $F$ into ${\aleph_0}$ non-empty sets.
• Take any partition $\mathcal{F}_{2}$ of $F$ into $2$ non-empty sets.
• Define for $i \in \{2, \aleph_0, 2^{\aleph_0}, 2^{2^{\aleph_0}}\}$ relation $\sim_i$ as $$f \sim_{i} g \iff \exists A \in \mathcal{F}_i.\ f \in A \land g \in A.$$

I hope this helps $\ddot\smile$