I have no idea, how to solve this problem:
Let $\psi: (\mathbb{R} \to \mathbb{N}) \to P(P(\mathbb{R}))$ defined as:
$\psi (f) = \mathbb{R}/_{ker(f)}$
(1) Is this a surjective function?
(2)Is this an injective function?
I tried to do something, but after hours I can't see any progress... I just don't know, were/how to start solving such problem. And I'm not sure, if I properly understand problem itself; I'm sure, that your explanation woluld be helpful.