Let p and q by any two numbers in N.

Let R be the set of numbers produced by all combinations of p/q.

As x approaches infinity, what does |R| approach?

The size(N) is obviously x.

The size(R) has the upper bound <= x

^{2}, as there are x

^{2}combinations of p and q, but there are redundancies ( ex 1/2 and 2/4), so we should be able to get a better upper bound.

size(R) has the lower bound >=x, because every integer is a rational.

Another lower bound would be 2(π(x)^2-π(x)) (π(x) being the prime counting function), as each combination of different primes can be guaranteed to be a unique rational, as can it's inverse.