I´m currently working on the following problem:
"Let $\xi$ = $\{ $ $\bot$ $\mid $$\bot$ is a equivalence relation over $\mathbb{N} $$\} $
Show that $\xi$ and $2^\mathbb{N} $ (power set) are equinumerous."
Edit: After some research, I found that I need to proof that $2^\mathbb{N} $ is equinumerous with $2^\mathbb{NxN} $, because $\xi $ $\space{ }$ $\subseteq$ $\space{ }$$2^\mathbb{NxN} $. Any ideas?
To inject $2^{\Bbb N}$ into $\xi$ (to within a factor $2$, but that is finite and can be patched up), given a subset you have an equivalence relation by the partition that subset/its complement.
To show that $|\xi| \le |\Bbb R|$, if $A_i$ is an element of the partition, associate with it the real number $a_i=\sum_{j \in A_i}3^{-j}$. Now associate the whole partition with the real $\sum_{i=1}^{|A|} a_i4^{-i}$