Thread: Binary operations, groupoids

I have the following problem:

Let (M; *) be a groupoid in which the following relations hold:
i) x * x = x; x belongs to M
ii) (x * y) * z = (y * z) * x; x, y, z belong to M.
Show that the operation * is commutative.

I don't know how to start to prove this. Please help Much appreciated!

Let x,y be in M. Then

x*y = x*(y*y) (using (i))

= (x*y)*y (associativity)

= (y*y)*x (using (ii))

= y*x (using (i).