math problem

This isn't homework, I'm just wondering if it's possible:

Construct a bijective map, f: N -> X, such that X is a countably infinite pairwise disjoint collection of countably infinite sets of rational numbers.

I can't find any good reason why this shouldn't work, but I can't come up with anything either.

Let f(X)=X
f(N)=N
Done

Make a bijection from N to the rationals, make a bijection from N to your collection and then compose them.

I said pairwise disjoint

list your rationals. then arrange them into sets {first},{second},....

then construct set X defined as {{first},{second},...} this is obviously countable. since its countable you have a bijection from N to X

OP here, i figured it out. f(n) = {p, p^2, p^3,...}, where p is the nth prime number.

i thought it would be more exciting :(