Let N be the set of positive integers. If A,B,C=∅, A∩B=B∩C=C∩A=∅ and A∪B∪C=N, we say that A,B,C are partitions of N. Prove that there are no partitions of N,A,B,C, that satis fy the following:
(i) ∀a∈A,b∈B, we have a+b+1∈C
(ii) ∀b∈B,c∈C, we have b+c+1∈A
(iii) ∀c∈C,a∈A, we have c+a+1∈B