Farey sequence and Graham's conjectures

Let Fn be the Farey sequence of order n. For S?Fn we let Q(S)={x/y:x,y?S,x?y and y?0}. We show that if Q(S)?Fn, then |S|?n+1. Moreover, prove in any following cases: (1) Q(S)=Fn; (2) Q(S)?Fn |S|=n+1, must have S={0,1,12,…,1n} or S={0,1,1n,…,n?1n} except for n=4, where an additional set {0,1,12,13,23} second case. Our results are based on Graham's GCD conjectures, which been proved by Balasubramanian Soundararajan.