Let S1, S2, S3,... be the sequence of sets defined by S1 = {1}, S2 = {2} and Si = Si-2 U (Si-1 Δ {1, 2, ..., i}) for each integer i ≥ 3. Prove by strong induction that, for all integers n ≥ 1, Sn = {1, 3, 5, ..., n} if n is odd and Sn = {2, 4, 6, ..., n} if n is even. Hint: Consider splitting your induction step into two cases: one for when k is even and the other for when k is odd.