Cardinality of equivalence relations on N I asked a similar question yesterday about well ordered sets, now I am having troubles with equivalence relations. 
Could someone suggest an injection from a well known set of cardinality
$2^{\aleph_{0}}$ to the set of all equivalence relations of $\mathbb{N}$?
Many thanks in advance!

For each subset $S$ of $\mathbb{N}$, declare all elements of $S$ to be equivalent, and all elements outside $S$ to only be equivalent to themselves. It's not quite an injection, but if you ignore singleton and empty $S$, then it is.
And because there are uncountably many subsets of $\mathbb{N}$, removing countably many won't hurt, and so the domain of this map is uncountable.
