I have the following problem: prove that the integer solutions to $x^2 + y^2 = z^2$ with $(x, y, z) = 1$ are in 1-to-1 correspondence with the rational solutions to $u^2 + v^2 = 1$.
Now, I have obtained an integer parametrization of the Diophantine equation with $x=2rs, y = r^2 + s^2$ and $z = r^2 - s^2$. Moving things around in the Pythagorean theorem we find that we can take $u = \frac{2rs}{r^2 + s^2}$ and $v = \frac{r^2+s^2}{r^2 - s^2}$ to obtain $u^2 + v^2=1$ with rational solutions.
My question is this: am I done? Is it enough to show that we can go from one solution to the other? I feel like the answer is no and I should produce an isomorphism between both set of equations. But then I thought, being isomorphic is the same as having the same cardinality and since both set of solutions are countably infinite, by the definition of cardinality we would be done right?
Thanks for the help.