Exercise 23, § 7

Let F be a free group with basic family $(x, y), x \neq y$. Show that the derived group of F has as basic family the family of commutators $(x^i, y^j), i \in \mathbf{Z}, j \in \mathbf{Z}, i \neq 0, j \neq 0.$ (Use Exercise 20 or Exercise 32.)