suppose $F$ is a free group generated by $x$ and $y$.
- Prove that $u=x^2, v=y^3$ generates a subgroup of $F$, and it is isomorphic to the free group on $u,v$
- Prove that $u=x^2, v=y^2, z = xy$ generates a subgroup of $F$, and it is isomorphic to the free group on $u,v,z$
My problem is below: I think what they want me to prove is the same thing in my mind, because I think, for example in problem 1, the two groups they mentioned are just a same group generated by $u$ and $v$. What should I prove and how?