You and your two friends, Tara and Peter, are having a discussion about positive integers, as you do.
Tara: I have divided all the positive integers into two non-empty disjoint sets A and B. Peter: Big deal. So did I, but I called mine S and T. You: Tara, what’s so special about your partitions? Tara: Well, a number is in A if and only if it’s the sum of 2 distinct integers in A or it’s the sum of 2 distinct integers in B.
You: What about yours, Peter? Peter: The sum of any two distinct integers in S is in T, and the sum of any two distinct integers in T is in S.
You: I don’t think either of these is possible!
Who’s right? Could Tara produce such a partition? What about Peter? For each, find solutions if possible; if not possible, explain why not.
Neatly written or typed submissions may be emailed to Professor Jonathan Bloom (firstname.lastname@example.org). Submissions are due by 6 a.m. on Saturday, March 7th.