Let S be a set of 2006 numbers. We call a subset T of S naughty if for any two arbitrary numbers u, v (not neccesary distinct) in T, u+v is not in T. Prove that
1) If S={1,2,…,2006} every naughty subset of S has at most 1003 elements;
2) If S is a set of 2006 arbitrary positive integers, there exists a naughty subset of S which has 669 elements. number theoryalgebra unsolvedalgebra